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# Create a new infix operator [duplicate]

It is well known that the infix operator @@ is used as a short form of Apply.

For example, both Apply[f,{a,b,c}] and f@@{a,b,c} return f[a,b,c]}

Also the infix operator @@@ is used as a short form of Apply when it is used for levelspec {1}

For example, both Apply[f,{{a,b},{c,d}},{1}] and f@@@{{a,b},{c,d}} return {f[a,b],f[c,d]}

I would like to create a new infix operator named @@@@ (or other) to express Apply[f,expr,{2}] in a short form like f @@@@ expr.

Is it possible to create a new infix operator equivalent to some more verbose command ? Also which are the valid names I could use ?


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### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# (a) Show that a polynomial of degree $$3$$ has at most three real roots.(b) Show that a polynomial of degree $$n$$ has at most three real roots.

(a) The given equation has at most three real roots.

(b) A polynomial of degree $$n$$ has at most $$n$$ real roots.

See the step by step solution

## Step 1: Given data

A polynomial of degree $$3$$ and $$n$$.

## Step 2: Concept of Rolle’s Theorem

“Let a function $$f$$ satisfies the following conditions,

1. A function $$f$$ is continuous on the closed interval $$(a\;,\;b)$$.

2. A function $$f$$ is differentiable on the open interval $$(a\;,\;b)$$.

3. $$f(a) = f(b)$$

Then, there is a number $$c$$ in open interval $$(a\;,\;b)$$ such that $${f^\prime }(c) = 0$$.”

## Step 3: Calculation to show that polynomial of degree $$3$$ has at most three real roots

(a)

Let the polynomial of degree $$3$$ as $$P(x)$$.

Suppose, $$P(x)$$ has four distinct real roots and $${b_1},{b_2},{b_3}$$ and $${b_4}$$ where. $${b_1} < {b_2} < {b_3} < {b_4}$$.

Then, $$f\left( {{b_1}} \right) = f\left( {{b_2}} \right) = f\left( {{b_3}} \right) = f\left( {{b_4}} \right) = 0.$$

Since the polynomial is continuous on $$(a,b)$$ and differentiable on $$(a,b)$$, then Rolle's Theorem implies that there is a number $${c_1},{c_2}$$ and $${c_3}$$ with $${b_1} < {c_1} < {b_2},{b_2} < {c_2} < {b_3}$$ and $${b_3} < {c_3} < {b_4}$$ in $$(a,b)$$ such that $${P^\prime }\left( {{c_1}} \right) = {P^\prime }\left( {{c_2}} \right) = {P^\prime }\left( {{c_3}} \right) = 0$$.

So, $${P^\prime }(x) = 0$$ must have at least three real solutions.

Thus, the derivative of $$P(x)$$ is a polynomial with degree 2.

It must have two roots but this contradicts the fact that $${f^\prime }(x) = 0$$ have at least three real solutions.

Thus, it can be concluded that $$P(x)$$ cannot have four real roots.

Hence, the given equation has at most three real roots.

## Step 4: Calculation to show that polynomial of degree $$n$$ has at most three real roots

(b)

Let the polynomial with degree $$n$$ be $$P(x)$$.

Prove this by induction method.

When $$n = 1$$, it is obvious that a polynomial of degree 1, has at most one root.

When $$n = k$$, assume that a polynomial of degree $$k$$ has at most $$k$$ roots.

When $$n = k + 1$$, to prove a polynomial of degree $$k + 1$$ has at most $$k + 1$$ roots.

Let the polynomial $$P(x)$$ with degree $$k + 1$$.

It is clear that $${P^\prime }(x)$$ is the derivative of $$P(x)$$ has degree $$k$$.

Assume that $$P(x)$$ has $$k + 2$$ distinct real roots and let the roots are $${a_1},{a_2}, \ldots ,{a_{k + 2}}$$ then, $$P\left( {{a_1}} \right) = P\left( {{a_2}} \right) = \cdots = P\left( {{a_{k + 1}}} \right) = P\left( {{a_{k + 2}}} \right) = 0$$.

Since the polynomial is continuous on $$(a,b)$$ and differentiable on $$(a,b)$$, then Rolle's Theorem implies that there is a number $${c_1},{c_2}, \ldots ,{c_{n + 1}}$$ with $${a_1} < {c_1} < {a_2},{a_2} < {c_2} < \cdots < {a_{k + 1}} < {c_{k + 1}} < {a_{k + 2}}$$ in $$(a,b)$$ such that $${P^\prime }\left( {{c_1}} \right) = {P^\prime }\left( {{c_2}} \right) = \cdots = {P^\prime }\left( {{c_{k + 1}}} \right) = {P^\prime }\left( {{c_{k + 2}}} \right) = 0$$.

So, $${P^\prime }(x) = 0$$ must have at least $$k + 2$$ real solutions.

Thus, the derivative of $$P(x)$$ is a polynomial with degree $$k + 1$$.

It must have $$k + 1$$ roots but this contradicts the fact that $${f^\prime }(x) = 0$$ must have at least $$k + 2$$ real solutions.

Thus, it can be concluded that the given equation cannot have at least $$k + 2$$ real roots rather it has at least $$k + 1$$ real roots.

Hence, a polynomial of degree $$n$$ has at most $$n$$ real roots.

format: plain


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# Time evolution of a discrete 1-d lattice of spin-(1/2) particles under a given Hamiltonian, or special cases thereof

I am trying to get some feel for the dynamics induced on a discrete 1-d lattice of spin-(1/2) quantum particles by the following Hamiltonian

$\hat{H} = \sum_{i, j} r_{i j} \left[ \mathrm{sin}(\alpha_{i j}) \left(\hat{S}^{x}_{i} \hat{S}^{x}_{j} - \hat{S}^{y}_{i} \hat{S}^{y}_{j} \right) + \mathrm{cos}(\alpha_{i j}) \left(\hat{S}^{x}_{i} \hat{S}^{y}_{j} + \hat{S}^{y}_{i} \hat{S}^{x}_{j} \right) \right]$

where $i$ and $j$ are particle labels, $r_{i, j}$ quantifies the overall interaction strength between particles $i$ and $j$, $\alpha_{i j}$ quantifies the relative strength of the two terms for these particles, and $\hat{S}^{\{x, y\}}_i$ are the $x$ and $y$ spin-(1/2) angular momentum operators on particle $i$.

Can anyone point me to some literature on this family of Hamiltonians or, if anyone recognizes this as a general case of a more well-characterized family, some literature on that special case? Thank you in advance!

• Do you know anything about $r_{ij}$? (If not, why is it a 1D lattice?) – Norbert Schuch Feb 26 '15 at 18:58
• Actually, that's sort of what I am asking. What constraints can I put on the $r_{ij}, \alpha_{ij}$ (e.g. translational invariance) to arrive at a well-characterized model? Sorry for making that unclear. – Adrian Feb 26 '15 at 20:57
• This makes this a rather broad question. If you only have nearest neighbor interactions, this is can be transformed to the XX model (or directly solved by mapping it to free fermions). If the $r_{ij}$ only act between even and odd sites, you can transform this to a model of hopping particles (but this can be hard to solve). --- In any case: Why do you call this a 1-d lattice? The way you write it there is no 1-d structure. – Norbert Schuch Feb 26 '15 at 21:35


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# Comparing two recurrence relations w.r.t. asymptotic growth

I have two functions $T_1(n),T_2(n)$. How do I decide which is asymptotically faster?

One is given by the recurrence relation

$$T_1(n) = \sqrt{n} T_1(\sqrt{n}) + 3 n, \quad T_1(1) = T_1(2) = 1.$$

The other is given by the recurrence relation

$$T_2(n) = 3 T_2(n/3) + 2n \log n, \quad T_2(1) = T_2(2) = 1.$$

For the first function I guess there is $O(\sqrt{n} \cdot \sqrt{n})$ for loop, and $O(n)$ for the $c$; which becomes $O(n^2)$ in total.

For the second one, the Master's theorem is applicable, but as I assume the complexity becomes $O(n \cdot n \log n) \Leftrightarrow O(n^2 \cdot \log n) \Rightarrow O(n)$ for loop, and $O(n\log n)$ for $c$.

So if I am comparing both $O()$'s, we can in total see that

$$O(n^2) < O(n^2 \cdot \log n)$$

Was there any mistake, or is this not how recurrence unrolling is done?

• What do you mean by "$O(n^2)<O(n^2\log n)$? Note that, for example, $n^2\in O(n^2)$ and $n\in O(n^2\log n)$ but $n^2>n$ for $n>1$. Nov 4 '16 at 10:49
• I have two questions? 1. Does my assumption that first functon is faster is correct? 2. Have i made correct O's from the recurrance functions? Nov 4 '16 at 11:43
• The answers to This question should provide an answer to your question about $T_1$. Nov 4 '16 at 13:42
• Terminology: the notion of "faster" doesn't apply to functions. To be precise you should either ask "which of these functions is asymptotically smaller?" or "These are timing functions for two algorithms, which algorithm is asymptotically faster?" Nov 4 '16 at 15:13
• – Raphael
Nov 7 '16 at 10:15

You can sort of get a feel of the asymptotic behavior of a recurrence by flatting out the recurrence into an equivalent problem of the form $A_k = c_1(k) A_{k-1} + c_2(k)$.
First of all, to see that $O(n^2)$ is an extremely loose bound, notice how $T_1(16)$ expands out: \begin{align*} T_1(16) &= 4 T_1(4) + 3 \cdot 16 \\ &= 4 (2 T_1(2) + 3 \cdot 4) + 3 \cdot 16 \\ &= 8 + 2 \times (3 \cdot 16) \end{align*} Similarly, you'll find that $T_1(256)$ expands out to \begin{align*} T_1(256) &= 16 T_1(16) + 3 \cdot 256 \\ &= 16 (8 + 2 \times (3 \cdot 16)) + 3 \cdot 256 \\ &= \frac{256}{2} + 3 \times (3 \cdot 256) \end{align*} This strongly suggests that $T_1(2^{2^k}) = 2^{2^{k-1}} + 3k 2^{2^k}$, or $T_1(n) \approx \frac{n}{2} + 3 n\log_2 \log_2 n$. We can show this by first relaxing the domain of the function to those of the form $2^{2^k}$. Since $\sqrt{2^{2^k}} = 2^{2^{k-1}}$, we have $$T_1(2^{2^k}) = 2^{2^{k-1}} T_1(2^{2^{k-1}}) + 3 \cdot 2^{2^k}$$ Let $A_k = T_1(2^{2^k})$, then \begin{align*} A_k &= 2^{2^{k-1}} A_{k-1} + 3 \cdot 2^{2^{k}} \\ &= 3 \cdot 2^{2^{k}} + 2^{2^{k-1}}(3 \cdot 2^{2^{k - 1}} + 2^{2^{k - 2}}A_{k-2}) \\ & \cdots \\ &= 2^{2^{k - 1}}A_{0} + 3 \cdot 2^{2^{k}} + 3 \cdot 2^{2^{k}} + \stackrel{k}{\dots} + 3 \cdot 2^{2^{k}} \end{align*}
which shows that when $n = 2^{2^k}$, then $T_1(n) = \Theta(n\log\log n)$. Suppose we now open $T_1(n)$ over the whole real domain (that is, we relax the domain restriction imposed previously), then it should be easy to show that if $T_1(x) = 1$ whenever $x \in [1, 2]$, then $T_1(n) = \frac{n}{2} + 3n \log_2\log_2 n$ extends uniformly. (Proof: Use the same derivation above, but for some other $p \in [1, 2]$, do the case analysis on $p^{2^k}$ instead)
Therefore, we have $T_1(n) = \Theta(n \log\log n)$ and $T_2(n) = \Theta(n^2 \log n)$. Asymptotically, $T_2$ still dominates $T_1$.
• Btw, the Master's theorem gives - $T_2(n) = \Theta(n\log^2 n)$ Nov 8 '16 at 10:14


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Viewpoint
Enthusiast

## QCD with no index

Hi,

Can you please provide me scenarios where I can have QCDs with no index, excepting that I exclude index subtable?

Thanks and regards,

Raja

2 REPLIES

## Re: QCD with no index

Raja,

I don't see how your question relates to Viewpoint. This would be better under the Database forum. There is a QCD thread there already:

Suggest you re-post something there.

Enthusiast

## Re: QCD with no index

Hi Gary,

Thanks for direction.

Regards,

Raja


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# Substitution of expressions for the output

I have a code which gives this kind of output :

W = Sum[v[j]*b[j], {j, 1, 3}];
AA = (v[1]*b[1] + v[2]*b[2] + v[3]*b[3])*2 + 3 ;

I know that if i want to substitute, for example, v[1] with 2 I have to do:

AA = (v[1]*b[1] + v[2]*b[2] + v[3]*b[3])*2 + 3  /. v[1] -> 2

But if i want to substitute the expression of W how should i do? Because if i do this:

    W = Sum[v[j]*b[j], {j, 1, 3}];
AA = (v[1]*b[1] + v[2]*b[2] + v[3]*b[3])*2 + 3  /.
W -> (b[1] v[1] + b[2] v[2] + b[3] v[3])

I unfortunately obtain:

3 + 2 (b[1] v[1] + b[2] v[2] + b[3] v[3])

Simplify[AA, W == ww]


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# Simplest model of chaos with time-independent smooth Hamiltonian and trivial topology?

What is the simplest model of chaos governed by a time-independent smooth Hamiltonian on a phase-space with trivial topology?

We know that...

• With trivial topology, the minimal number of dimension to exhibit chaos with first-order ODEs is three (e.g., the Lorentz system), but these are of course not Hamiltonian.

• Indeed, time-independent Hamiltonian systems of two phase-space dimensions (one configuration variable) cannot be chaotic because the trajectories follow the fixed-energy surfaces which foiliate the phase space. Thus we need four phase-space dimensions (two configuration variables).

• There exist time-dependent examples of Hamiltonian chaos in two-dimensional phase space, e.g., the kicked-top.

• There exist discrete-time non-smooth chaotic maps in two-dimensions that preserve area, e.g., Baker's map.

• Hadamard's billiards is a time-independent chaotic Hamiltonian system on a four-dimensional phase space. However, it has non-trivial topology (a two-holed donut), exhibiting chaos on account of constant negative curvature.

The double pendulum with equal masses and equal arm lengths has Hamiltonian

$$H(\theta_1,p_1,\theta_2,p_2)= \frac{1}{6} m l^2 \left ( {\dot \theta_2}^2 + 4 {\dot \theta_1}^2 + 3 {\dot \theta_1} {\dot \theta_2} \cos (\theta_1-\theta_2) \right ) - \frac{1}{2} m g l \left ( 3 \cos \theta_1 + \cos \theta_2 \right ).$$

where $\theta_1$ and $\theta_2$ are the angles of the top and bottom arm with respect to the vertical direction, and $p_1$ and $p_2$ are the respective conjugate momenta. This satisfies all our specific requirements except that it is not very simple.

(Note that I am merely extending this question on /r/physics, which lacked the specialization to topologically trivial phase spaces.)

1. A 1D autonomous system is always Liouville integrable – the Hamiltonian $H$ itself is an integral of motion – so we would have to consider at least 2D to find chaos.

2. A quadratic potential $V$ yields a linear system, which doesn't exhibit chaos. So the potential $V$ should contain cubic or higher terms.

3. The potential should mix coordinates (e.g., terms like $x_2^2x_1$) so that the system isn't separable into two 1D systems.

4. The Henon-Heiles system in 2D with a potential $V$ containing both quadratic and cubic terms is often given as the standard example of chaos. See e.g. this Phys.SE post.

5. The homogeneous Henon-Heiles system with a purely cubic potential $V$ is an even simpler system: the Hamiltonian $$H(x_1,p_1,x_2,p_2) = \frac{g}{3}x_1^3 + x_1 x_2^2 + \frac{1}{2}\left(p_1^2 + p_2^2\right)$$ is non-integrable for many value of $g$, including $g \in (-\infty,1)$. See Example 3 (Eq. 5.2) in Section 5.1 of Ref. 1.

References:

1. J.J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Math. 179 (1999). [free PDF].
• This is darn close to being an objective answer to "simplest", a seemingly subjective criterion. Just what I was hoping for. Stupendous! Thanks Qmechanic for showing there's still a bit of magic on Physics.SE. – Jess Riedel May 18 '18 at 18:47
• Indeed, pretty good answer! Note that, besides the citeseerx link provided by @JessRiedel 's edit, a pdf file of Ref. 1 is also available in ResearchGate. – stafusa May 18 '18 at 19:19


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## Introduction

As previously discussed by Richard P. Feynman, a negative probability, which relaxes the axiom of a non-negative probability of an event in Kolmogorov’s probability theory, sheds new light on our understanding of quantum phenomena1,2. The essence of his idea is that a negative probability results in much less mathematical complications in intermediate steps for the analysis of a given physical event. As an example, Feynman developed joint probability distributions for spin-1/2 systems to address Young’s double-slit experiment using a different approach, such that the probability distributions can have negative values1. Such an idea has been applied to many studies involving various quantum phenomena3,4,5,6,7. In particular, the negative probability approach provides new insight into a disagreement between classical and quantum predictions on the Bell’s theorem8. It is experimentally verified that our nature cannot be described by any classical theory of local realism, as the classical theory obeys but the quantum theory violates the Bell inequalities9,10,11. Negative probability introduces a different point of view that local realistic theory endowed with a negative probability can simulate violations of the Bell inequalities, similar to quantum theory12,13,14. However, these negative probabilities are distinct from the definition of probability with respect to the relative frequency of events, in which case, an operational interpretation is not necessarily straightforward (for historical review, see ref. 15).

On the other hand, negative values play a role as an indicator. Let us retrace Feynman’s example on the trading of apples1: “A man starting a day with five apples who gives away ten and is given eight during the day has three left.” Feynman pointed out that the initial and final numbers of apples, we denote by (5,3) and call a fundamental entity, can be calculated in the following two ways. The first case reads “beginning with 5 apples at time $${t}_{1}$$, given 8 at $${t}_{2}$$, giving away 10 at $${t}_{3}$$, and having 3 left at $${t}_{4}$$”, or symbolically, $${5}_{{t}_{1}}+{8}_{{t}_{2}}-1{0}_{{t}_{3}}={3}_{{t}_{4}}$$ with a time ordering $${t}_{i} < {t}_{j}$$ for $$i < j$$. This can be claimed to be natural since the apple number is always kept as non-negative. On the other hand, Feynman introduced an alternative model which disregards keeping the apple number as nonnegative but maintains the same fundamental entity $$({5}_{{t}_{1}},{3}_{{t}_{4}})$$; it reads $$5-10=-\,5$$, and $$-5+8=3$$. Note that the latter model disregards the chronological order $${t}_{i}$$, i.e., changes the order of trading, thus a negative number of apples appears in the intermediate steps. Although such negative numbers are abstract, allowing negativity leads to have more freedom in mathematical analysis without altering the fundamental entity. We say that such a model is free of time context16,17. In other words, the time context-free approach maintains the same fundamental entity at the expense of relaxing the assumption of nonnegative apples. Moreover, the negative apples in the intermediate steps introduce the idea of “debts” with respect to the trading of apples (the indicator).

In quantum optics, negative values have been widely used as an indicator of nonclassicality in regard to classical statistics. For instance, the Wigner function, one of the so-called quasiprobability distributions, is used to represent a joint distribution of the position and momentum in phase space18. However, due to the uncertainty principle with such conjugate variables, the Wigner function is negatively valued for some quantum states. It witnesses quantum phenomena of the states and has been generalized to finite dimensional systems such as quantum bits, and currently the quasiprobability approach has been applied to the omnidirectional range of fields in quantum information science19,20,21,22,23,24,25,26,27,28,29,30.

However, such nonclassicality indicators with Wigner functions lack the operational formalism wherein preparation, operation, and measurement cooperate explicitly16,17. Negative values in one quasiprobability can be positive in another. This fact can be an obstacle to the operational interpretation of these negative values. Furthermore, their comparison to classical statistics reveals the subtlety that quasiprobabilities require different physical interpretations for the same form of functionals as their classical counterparts. This is described as “incommensurability” of quasiprobabilities31,32. Therefore, it is more natural to employ quasiprobability that compares quantum and classical statistics on the same footing.

The operational quasiprobability (OQ) introduced in31,32 allows the problems in question to be resolved. The OQ is commensurate since it evaluates the statistics of the mathematical functionals with the same physical interpretation in every model, regardless of whether a quantum or classical case is being considered. The OQ consists of selective and sequential measurements in time and is formulated as joint probability distributions that simultaneously describe multiple measurement setups. We consider the expectation values of multiple measurement setups as fundamental entities, i.e., the measurable quantities of interest. The OQ method allows the joint distributions to be independent of the measurement setups and to simultaneously describe the multiple-setup outcomes, even though the distributions can be negative. This method can be considered as an alternative way of describing quantum theory that depends on the setups. It allows a direct comparison between quantum and classical statistics and identifies nonclassicality in an operational way.

We are focused on the specific feature that the moments will vary depending on the measurement(s) that are performed. This is called measurement-selection context and is similar to the time context of the Feynman’s apple example. The fundamental entity is a set of all moments in the single measurements. For two dichotomic observables, it is $$\{(\langle {A}^{n}\rangle ,\langle {B}^{m}\rangle )\}$$ with $$\langle {A}^{n}\rangle$$ being $$n$$-th moments in one of the single measurements and similarly $$\langle {B}^{m}\rangle$$ in the other. The OQ is free of the measurement-selection context, in the sense that the local marginals of the joint distribution are equal to the probabilities of the single measurements. This is one of the most astonishing features that classical models presume, including a macrorealistic model33. Moreover, the context-free OQ can always be constructed in quantum theory, even though quantum theory is not context-free. Instead, the quantum OQ pays a tariff of negative probabilities. Such inevitable negatives can be understood as an indicator of nonclassicality in the context of measurement selection.

The macrorealistic model assumes no-signaling in time (NSIT) and arrow of time (AoT); NSIT implies that a later measurement produces a result which is not affected by whether or not an earlier measurement is performed, and AoT is a similar condition where the role of later and earlier measurements is exchanged. In other words, roughly speaking, the two measurements are independent, and both of the sequential measurement and the individual measurements leave the fundamental entity unchanged. The classical model is thus free of the measurement-selection context. More explicitly, when two measurements are sequentially performed at times $${t}_{1}$$ and $${t}_{2}$$ with $${t}_{1} < {t}_{2}$$, respectively, or they are individually performed, the NSIT and AoT are described by

$$\begin{array}{rcl}NSIT:\ {P}_{{t}_{2}}({a}_{2}) & = & {P}_{{t}_{1},{t}_{2}}({a}_{2}),\\ AoT:\ {P}_{{t}_{1}}({a}_{1}) & = & {P}_{{t}_{1},{t}_{2}}({a}_{1}).\end{array}$$

Here $${P}_{{t}_{i}}({a}_{i})$$ are the probabilities of the single measurements, whereas $${P}_{{t}_{1},{t}_{2}}({a}_{i})$$ are the marginals of the joint measurement $${P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})$$ such that $${P}_{{t}_{1},{t}_{2}}({a}_{i})={\sum }_{{a}_{j\ne i}}{P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})$$ for $$i,j=1,2$$. In contrast, quantum theory violates the macrorealistic assumptions and it is contextual with the measurement selection.

However, its OQ representation allows the joint distribution to be free of measurement-selection context as in the macrorealistic model, and it pays the quantum tariff of negative probabilities. The OQ representation states that a quasiprobability distribution $${\mathscr{W}}$$ is operationally defined for both quantum and classical models by31,32:

$$\begin{array}{c}{\mathscr{W}}({a}_{1},{a}_{2})={P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})+\frac{1}{2}[{P}_{{t}_{1}}({a}_{1})-{P}_{{t}_{1},{t}_{2}}({a}_{1})]+\frac{1}{2}[{P}_{{t}_{2}}({a}_{2})-{P}_{{t}_{1},{t}_{2}}({a}_{2})].\end{array}$$
(1)

The $${\mathscr{W}}$$ has marginal probabilities equal to those of the single measurements. The presence of negative $${\mathscr{W}}({a}_{1},{a}_{2})$$ originates from the statistical difference between the single and sequential measurements. Note that $${\mathscr{W}}$$ will be nonnegative joint probabilities if NSIT and AoT are applicable. Thus, measuring OQ represents an evaluation of the context-free model for a given experiment on one hand, and also is a test of whether a given system violates the macrorealistic model based on its negative values on the other hand. Recently, experiments to test macrorealism have been conducted with various quantum states and measurement schemes34,35,36,37.

It is worth noting that a negative OQ suffices for the failure of the macrorealistic description but is not necessary. Despite the violation of the assumptions, we can still observe a positive OQ depending on the statistical differences. Such a behaviour was reported by employing general measurements32. This implies that the macrorealism, or the conditions of NSIT and AoT cannot fully capture the classicality reflected by the OQ positivity in the context of measurement-selection.

We now experimentally illustrate the negative probability in an operational way by measuring the degrees of freedom of the polarization of single photons. By considering the negative quasiprobability together with the quantum nature of photons, nonclassicality was demonstrated in the laboratory. To this end, we selectively implemented two sets of polarizations at consecutive times; the horizontal/vertical (H/V) measurement at $${t}_{1}$$ and the diagonal/anti-diagonal (D/A) at $${t}_{2}$$. Let us denote the selection of measurements by a tuple $$({n}_{1},{n}_{2})$$. We can then perform the following four measurement setups: (i) no measurement by $$({n}_{1},{n}_{2})=(0,0)$$, (ii) H/V single measurement by $$({n}_{1},{n}_{2})=(1,0)$$, (iii) D/A single measurement by $$({n}_{1},{n}_{2})=(0,1)$$, and (iv) consecutive joint measurement of H/V and D/A by $$({n}_{1},{n}_{2})=(1,1)$$. In this way, each of the probabilities at times $${t}_{i}$$ in equation 1 is associated with the tuple. We represent the experimental results by the notation of times for convenience. For example, the joint probabilities $${P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})$$ are equal to $${P}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})$$ with $$({n}_{1},{n}_{2})=(1,1)$$. The negativity is defined by the sum of the negative components of $${\mathscr{W}}({a}_{1},{a}_{2})$$: $${\mathscr{N}}\equiv \frac{1}{2}{\sum }_{{a}_{1},{a}_{2}}\left[\left|{\mathscr{W}}({a}_{1},{a}_{2})\right|-{\mathscr{W}}({a}_{1},{a}_{2})\right].$$ The maximum of the negativity is $$(\sqrt{2}-1)/4\approx 0.104$$ in case of two measurements31. We investigated the negativity of OQ for two input sources: (i) heralded single photons generated by spontaneous parametric down-conversion (SPDC) and (ii) single photons emitted from a single molecule. Note that all inputs were set to generate a single photon.

For the generation of heralded single photons, we exploited collinear type-II SPDC as shown in Fig. 1b. The signal was counted only when the trigger ($${D}_{trigger}$$) clicked. The input polarization state of a single photon is given by $$|\Psi (\theta ,\varphi )\rangle =\cos (\theta /2)|H\rangle +{e}^{i\varphi }\sin (\theta /2)|V\rangle$$ with $$\theta ,\varphi \in [0,9{0}^{\circ }]$$, where the angles of waveplates determine $$\theta$$ and $$\varphi$$ (see Methods). To experimentally implement the four measurement setups for OQ, which are described in Fig. 1a, we used the arrangement depicted in Fig. 1b. Three polarizing beam splitters (PBSs) and two half-wave plates (HWPs) are used to selectively measure the polarization states of photons. The PBS$${}_{1}$$ is used for the H/V polarization measurements and the two PBS$${}_{2}$$ and PBS$${}_{3}$$ with the HWPs are used for the D/A polarization measurements. Selective measurements can be performed by moving each PBS in and out of the path of the input beam. In the laboratory, we positioned two PBS$${}_{2}$$ and PBS$${}_{3}$$ at fixed positions to reduce experimental errors, and moved only the PBS$${}_{1}$$ to implement the four measurement setups. In the detection part, we counted the relative ratio of measurement outcomes as follows: $${P}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})={N}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})/{N}_{{n}_{1},{n}_{2}}^{tot}$$, where $${N}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})$$ denotes the sum of the counted photons at the detector $${D}_{{a}_{1},{a}_{2}}$$ and $${N}_{{n}_{1},{n}_{2}}^{tot}$$ is the total number of counted photons at all detectors for a given setup $$({n}_{1},{n}_{2})$$. See Methods for more details.

Figure 2

shows the negativity of OQ by the heralded single photon as a function of $$\theta$$ and $$\varphi$$ in the state $$|\Psi (\theta ,\varphi )\rangle$$. For $$\varphi ={0}^{\circ }$$, we observed the negativity for all range of $$\theta$$ (see Fig. 2a). The error bars are obtained by considering the functioning errors of the optics and devices used (see Supplementary Information). Figure 2b shows a contour plot of the negativity as varying the variables $$\theta$$ and $$\varphi$$. The maximum negativity yields $${\mathscr{N}}\approx 0.103$$ at $$\theta =4{5}^{\circ }$$ and $$\varphi ={0}^{\circ }$$, which well reproduces the theoretical prediction31. The negativity clearly indicates that any classical models that assume the NSIT and AoT conditions, cannot describe the nonclassicality of a single photon.

The quantum nature and the negativity of a single photon is reduced due to the decoherence. Figure 3a shows the theoretical values of the negativity on a cross-section of the Bloch sphere for $$\varphi ={0}^{\circ }$$. A pure state in the form of $$|\Psi (\theta ,\varphi =0)\rangle =\cos (\theta /2)|H\rangle +\sin (\theta /2)|V\rangle$$ with $$0\le \theta \le 36{0}^{\circ }$$ is placed on the rim of the plane. The inner points correspond to mixed states which are in the probabilistic mixture of the pure states; $$\hat{\varrho }({\theta }_{1},{\theta }_{2},\alpha )=\frac{1+\alpha }{2}|\Psi ({\theta }_{1})\rangle \langle \Psi ({\theta }_{1})|+\frac{1-\alpha }{2}|\Psi ({\theta }_{2})\rangle \langle \Psi ({\theta }_{2})|$$, where $$\alpha$$ determines the mixing ratio of two pure states with a constraint of $$\left|\alpha \right|\le 1$$. When $$\left|\alpha \right|=1$$, $$\hat{\varrho }$$ becomes the pure state. The central point of the circle indicates a completely depolarized state. The diamond dash lines in Fig. 3a represent the mixture of two pure states, where the negativity becomes zero and thus their convex region inside the dash lines also has a zero negativity. In this case, the experimental results, Fig. 3b, are again in well agreement with the theoretical predictions.

The heralded single photons exhibit an anti-bunching feature (see Supplementary Information). However, discussions on the second-order correlation function of these photons have been previously reported38,39,40. As a demonstration with a deterministic single photon source, we performed similar measurements with photons emitted from a single molecule (terrylene)41 (see Methods for more experimental details). In this case, the photon statistics clearly show the anti-bunching nature without any detection schemes such as triggering (see Supplementary Information). A similar negativity is obtained compared to the SPDC case (see Fig. 3c).

Finally, we also performed the same experiment with the weak-field light source (see Supplementary Information). Similar to the results obtained for the single photon sources, we also observed negative values. In this experiment, we post-selected the raw data to evaluate the negativity in a way that only single APD clicks were sampled and the rest of events, e.g., more than two clicks simultaneously were neglected. In general, the weak-field light is understood not to be the single-photon source in the sense that this light does not show the anti-bunching effects. However, the negativity can be detected with a post-selection process. Recently, such a phenomenon was reported that a coherent state of the optical field can show the nonclassicality32,42. We highlight that the operational quasiprobability reveals the negativity by an interplay between given state and measurement.

In conclusion, we experimentally explored the negativity of the operational quasiprobability by measuring the single photon polarizations. We introduced the context of measurement selection by constructing the quasiprobability such that by marginals it provides the same fundamental entity as that of the single measurements. As a result, the quasiprobability can reproduce the quantum predictions by allowing negative probabilities. The measured negatives highlight the discrepancy between the classical and quantum predictions in the context of measurement selection. In the case of the classical prediction, we investigated the macrorealistic model assuming the NSIT and AoT conditions. In this model, the operational quasiprobability becomes a legitimate joint probability distribution for the given measurement setups. Therefore, observing the negatives highlights the nonclassical property in the context of measurement selection. We note that negativity is merely a sufficient condition for violating the NSIT. That is, the operational quasiprobability can be nonnegative even if the NSIT is violated. Such a case is encountered if a general measurement is involved, which will be discussed in a forthcoming paper. From a fundamental perspective, the measured negativity provides an operational approach to unravel the nonclassicality of photons in the context of measurement selection.

## Methods

### Input preparation

Heralded single photons: The experimental schematic used to generate the heralded single photons is shown in Fig. 1b. Orthogonally polarized photon pairs are generated by a type-II SPDC process using a 401.5 nm continuous wave (CW) mode laser to irradiate a periodically poled KTiOPO$${}_{4}$$ (PPKTP). The resulting photon pair is separated using a PBS. The horizontally polarized photon (signal) is used as the input photon and the vertically polarized photon (idler) is used as the trigger. Simultaneous detection in the signal and idler channels exclude the contribution of the vacuum state of the SPDC source, and thus gives the anti-bunching property of the heralded single photons. We also experimentally examined this by measuring a second-order correlation function with multi-channel correlation measurements and obtained a value of $${g}^{2}(0)=0.036$$ (for more details, see Supplementary Information).

Single photons from a single molecule: The output of a CW-mode laser (532 nm) is focused using an oil objective (NA = 1.40) onto a single terrylene molecule which is embedded in a thin para-terphenyl crystal ($$\sim$$20 nm) in a total internal reflection geometry38. The emitted fluorescence signal transmitted across a long-pass filter (LPF) is collected by the same objective and diverted to the detection part. The measurement value of $${g}^{2}(0)$$ is 0.14 (see Supplementary Information).

Input polarization state: The input photon polarization state was set to a pure state for H/V polarization states in the form of $$|\Psi (\theta ,\varphi )\rangle =\cos (\theta /2)|H\rangle +{e}^{i\varphi }\sin (\theta /2)|V\rangle$$, where $$\theta$$ and $$\varphi$$ are the bases of the polar coordinates on a unit sphere, called a Bloch sphere. To prepare such a state, the horizontally polarized photons are sequentially transmitted through the half wave-plate (HWP) and two quarter wave plates (QWPs) as illustrated in Fig. 1b. The last QWP is fixed at the angle of $$\pi /4$$. The final polarization state of the input photon is obtained as follows:

$$\begin{array}{c}{T}_{QWP}(\frac{\pi }{4}){T}_{HWP}(p){T}_{QWP}(q)(\begin{array}{c}1\\ 0\end{array})={e}^{i(-2p+q+\pi /4)}(\begin{array}{c}\cos (\pi /4-q)\\ {e}^{i(4p-2q-\pi /2)}\sin (\pi /4-q)\end{array}),\end{array}$$
(2)

where $${T}_{Q(H)WP}$$ represents the transfer matrix of the corresponding wave-plate. The parameters of $$p,q$$, and $$\pi /4$$ are the rotating angles of the waveplates and follow the relations: $$p=(\pi +\varphi -\theta )/4$$ and $$q=(\pi /2-\theta )/2$$.

### Collecting data

In the detection part, we counted the relative ratio of measurement outcomes as follows: $${P}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})={N}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})/{N}_{{n}_{1},{n}_{2}}^{tot}$$, where $${N}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})$$ denotes the sum of the counted photons at the detector $${D}_{{a}_{1},{a}_{2}}$$ and $${N}_{{n}_{1},{n}_{2}}^{tot}$$ is the total number of counted photons at all detectors for a given setup $$({n}_{1},{n}_{2})$$. We here describe in detail how to obtain each component in Eq. 1 in terms of the measurement setting $$({n}_{1},{n}_{2})$$ and the detectors (APDs) $${D}_{{a}_{1},{a}_{2}}$$.

1. 1.

$${P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})$$: we implement the consecutive measurement of H/V and D/A, which corresponds to $${P}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})$$ with $$({n}_{1},{n}_{2})=(1,1)$$. Then, collect the data at all the APDs $${D}_{{a}_{1},{a}_{2}}$$.

2. 2.

$${P}_{{t}_{1}}({a}_{1})$$: this corresponds to the H/V single measurement that can be rewritten by $${P}_{{n}_{1},{n}_{2}}({a}_{1},0)$$ with $$({n}_{1},{n}_{2})=(1,0)$$. In laboratory, we fixed the PBS$${}_{2,3}$$, thus it can be obtained by the marginal of the $${P}_{{n}_{1},{n}_{2}}({a}_{1},{a}_{2})$$ with $$({n}_{1},{n}_{2})=(1,1)$$. That is, $${P}_{1,0}({a}_{1},0)={P}_{1,1}({a}_{1},0)+{P}_{1,1}({a}_{1},1)$$. In this case, the data is collected at $${D}_{{a}_{1},0}$$ for $${P}_{1,1}({a}_{1},0)$$ and $${D}_{{a}_{1},1}$$ for $${P}_{1,1}({a}_{1},1)$$.

3. 3.

$${P}_{{t}_{1},{t}_{2}}({a}_{1})$$: this is simply a marginal of $${a}_{2}$$ by $${P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})$$.

4. 4.

$${P}_{{t}_{2}}({a}_{2})$$: this corresponds to the D/A single measurement that is given by $${P}_{{n}_{1},{n}_{2}}(0,{a}_{2})$$ with $$({n}_{1},{n}_{2})=(0,1)$$. After the PBS$${}_{1}$$ is out, we collect the data at APDs $${D}_{0,{a}_{2}}$$.

5. 5.

$${P}_{{t}_{1},{t}_{2}}({a}_{2})$$: this is simply a marginal of $${a}_{1}$$ by $${P}_{{t}_{1},{t}_{2}}({a}_{1},{a}_{2})$$.

### Detection and data acquisition

The photon clicks of the single photon counting detectors (Perkin-Elmer, SPCM-AQ4C) are sent to a field-programmable gate array (FPGA, NI PXI-7841R) for the post-selection process. For data acquisition and processing, the FPGA operates at a 25 ns clock speed (40 MHz) and 125 ns processing cycle. Each detector has a slightly different detection efficiency; therefore, it is of crucial importance that their effective efficiencies are equalized. To this end, we measured the counts of the input polarization with the angle $$\theta =4{5}^{\circ }$$, where all four detectors are supposed to have the same input photon flux under the same detection efficiency. The detected counts are used as the references to normalize the signals of each detector.

The sum of the detected counts at four APDs is about $$1{0}^{4}$$ on average, for all values of $$\theta$$ and $$\varphi$$. Table 1 shows some examples of the detected counts for each APD from Fig. 2a.


text:
# Aircrafts

James make the 21 paper aircrafts. It is 9 pieces more than make Matthew. How many paper aircrafts maked Matthew?

Correct result:

M =  12

#### Solution:

$M = 21-9= 12$

We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

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text:
Contents

Bubble, Bubble, Flow and Hubble:

[.25cm] Large Scale Galaxy Flow from Cosmological Bubble Collisions

Klaus Larjo1 and Thomas S. Levi 2

Department of Physics and Astronomy

University of British Columbia

Center for Cosmology and Particle Physics

Department of Physics, New York University

4 Washington Place, New York, NY 10003.

We study large scale structure in the cosmology of Coleman-de Luccia bubble collisions. Within a set of controlled approximations we calculate the effects on galaxy motion seen from inside a bubble which has undergone such a collision. We find that generically bubble collisions lead to a coherent bulk flow of galaxies on some part of our sky, the details of which depend on the initial conditions of the collision and redshift to the galaxy in question. With other parameters held fixed the effects weaken as the amount of inflation inside our bubble grows, but can produce measurable flows past the number of efolds required to solve the flatness and horizon problems.

## 1 Introduction

One of the greatest challenges for string theory is to find a testable prediction which can differentiate it from other potential theories of quantum gravity and high energy physics. Unfortunately, the typical energy scales at which string theoretic effects become important is likely far out of reach for any earth-based particle accelerator. Cosmology though, provides another opportunity. The early universe was arbitrarily hot and dense and signatures of the physics at that time remain in the large-scale structure of the universe today.

It has become increasingly clear in recent years that one of the characteristic features of string theory is the “landscape” model of cosmology [1, 2, 3]. There are many ways to generate realistic four-dimensional universes in string theory via flux compactifications. Various parameters specifying the compactification become scalar fields in our four-dimensional universe. The landscape is a potential energy function of the would-be moduli scalar fields, which possesses many distinct minima. These minima are meta-stable and can decay via bubble nucleation-type tunneling or quantum thermal processes. In the early universe, one expects all the minima to be populated as the temperature decreases and the fields begin to settle. The minima with the highest values of potential energy then begin to inflate, and are expected to rapidly dominate the volume. After a few string times the universe will be almost completely full of rapidly inflating regions, and as time continues to pass the metastable field values in these regions will begin to decay by bubble nucleation. Since we live in a region with very small vacuum energy, we expect that our observable universe is contained in such a bubble.

The most promising way to test this idea is to look at large scale cosmological structure. If we model the string theory landscape as a scalar field coupled to gravity with a potential with several minima, universes inside a bubble which formed via a tunneling instanton have certain characteristic features: they are open, they are curvature dominated at the “big bang” (rendering it non-singular), and they have a characteristic spectrum of density perturbations generated during the inflationary period that takes place inside them [4, 5, 6, 7, 8, 9, 10].

Moreover, because the bubble forms inside a metastable region, other bubbles will appear in the space around it. If these bubbles appear close enough to ours, they will collide with it [11, 12, 13]. The two bubbles then “stick” together, separated by a domain wall, and the force of the collision releases a pulse of energy which propagates through each of the pair. This process (as seen from inside our bubble) makes the universe anisotropic, affects the cosmic microwave background and the formation of structure. A collision between two bubbles produces a spacetime which can be solved for exactly [14, 15, 16, 17].

In [15, 17], one of these authors began the study of this process and its effects. In these papers we principally studied the effects of such a collision on the cosmic microwave background (CMB). We found that bubble collisions naturally create hemispherical power asymmetries spread through the lower CMB multipoles and/or produce relatively small cold or hot spots, depending on the time of the collision (and hence the size of the disk), and can lead to measurable non-Gaussianities. The qualitative features, however, are always the same: a collision with a single other bubble produces anisotropies that depend only on one angle—the angle to the vector pointing from our location to the center of the other bubble—and which are generically monotonic in the angle.

While the CMB temperature map is a good probe of these effects, it is by no means the only one. In this paper, we begin to look for other measurable effects by studying how a single bubble collision affects large scale structure and galaxy motion. We find that collisions generically lead to a large scale coherent flow of galaxies in our sky. The size of the affected disk of galaxies, and the magnitude of the effect are controlled by similar order parameters for the CMB effect, as well as the redshifts of the galaxies in question. Generically, the angular size of the affected disk on the sky for large scale structure is always larger than the corresponding disk for the CMB hot/cold spots. It is very interesting to consider our results in light of recent observational measurements of bulk galaxy flows dubbed “dark flow” [18, 19, 20].

Our model consists of a collision between our bubble – which we treat as a thin-wall Coleman-de Luccia bubble which undergoes a period of inflation after it formed (which we model as a de Sitter phase) followed by a period of radiation domination – and another different bubble, the details of which are largely unimportant. We assume that the tension of the domain wall between the bubbles and the vacuum energy in the other bubble are such that the domain wall accelerates away from us. Such collisions can produce observable effects which are not in conflict with current data. We will also assume that the inflaton is the field that underwent the initial tunneling transition, so that the reheating surface is determined by the evolution of that field and can be affected by the collision with the other bubble through its non-linear equation of motion, and that no other scalar fields are relevant.

After some number of efolds of inflation, our bubble will reheat and become radiation dominated. The subsequent expansion and the geometry of the reheating surface itself will be modified by the presence of the domain wall. In [17], it was shown how the surface was modified. This modified surface will also backreact on the spacetime because the various fluids created at reheating will no longer be classically comoving. In [17] this backreaction was ignored and it was assumed the universe could be approximated as standard radiation dominated Friedmann-Robertson-Walker (FRW) universe for the subsequent evolution. We will show here that this approximation is valid for the CMB photons, but not for the matter fluid that will form the galaxies we are interested in here. To find the evolution of galaxies, some measure of this backreaction must be taken into account, and we make a first approximate attempt to do so here.

There are a number of anomalies in current cosmological data, including a cold spot [21, 22, 23, 24, 25], hemispherical power asymmetry [26], non-Gaussianities (see e.g. [27]), the “axis of evil” [28], and the tentative possibility of a “dark flow” [18, 19, 20]. While the significance of these effects is unclear, there are few if any first-principle models which can account for them. Our results here, combined with [15, 17] indicate that bubble collisions can naturally create hemispherical power asymmetries spread through the lower CMB multipoles, relatively small cold or hot spots, measurable non-Gaussianities and a possible macroscopic, coherent bulk flow of galaxies. While it is not yet clear that a single collision can produce all of these effects in agreement with observations, it is nevertheless interesting that bubble collisions can at least qualitatively produce many of these effects.

## 2 Basic Setup

In this section we will set up the basic scenario for the bubble collision and subsequent evolution we wish to study. The details of the dynamics of general collisions were studied in [15], and some effects on the CMB were studied in [17]. We refer the reader there for a more complete treatment. We will focus on a collision scenario where our bubble is dominated by positive vacuum energy at formation, so that it undergoes a period of inflation. We will further assume the pressures are such that domain wall that forms between the two bubbles is accelerating away from the center of our bubble at late times. This second assumption is valid for collisions between two dS bubbles if ours has a smaller cosmological constant.3 While more general scenarios are possible, these assumptions lead to an evolution that is consistent with current observations.

We study the motion of galaxies and large scale structure in this scenario. A full treatment would require solving the full cosmological perturbation theory and evolution taking into account the macroscopic anisotropy caused by the collision. This treatment is outside the realm of the current paper. Instead, we will make several assumptions so that an approximate analytic treatment is possible. Our basic scenario unfolds as follows: our bubble undergoes a period of inflation sufficient to solve the horizon, flatness and relic problems in line with current observations ( e-folds) after which the spacetime reheats. After reheating the universe can be approximately described by a radiation dominated (RD) Friedmann-Robertson-Walker (FRW) universe. We say approximately, because as shown in [17] the reheating surface is modified by the bubble collision and formation of the domain wall. In short, the domain wall causes the reheating surface to “bend” towards or away from the domain wall depending on the details of the collision. See Fig. 1 and [17] for more details. Because the universe now reheats along a deformed, non-comoving surface the spacetime is not truly RD-FRW, as some of the symmetry of this geometry is broken. In [17] we still approximated the subsequent evolution as taking place in RD-FRW. For CMB photons, this is a reasonable approximation (as will be shown later). However, for the matter fluid and galaxies, this approximation is no longer justified and some measure of the backreaction and modified geometry must be taken into account. We will attempt to do so here. In reality, the spacetime will also undergo a transition to matter domination at some later time after last scattering. We ignore this latter transition in this paper, as we believe the essence of the galaxies motion is captured in our setup. It would nevertheless be interesting to study this in more detail and see if our assumptions hold.

To find the approximate motion of galaxies, we assume that the initial conditions of the matter fluid are that it is moving orthogonal to the reheating surface. We will then attempt to both capture the backreaction of this fluid on the spacetime (thereby modifying our spacetime from RD-FRW) and track the motion of galaxies through this new spacetime by computing the timelike geodesics for each galaxy. We are thus making several assumptions, some of which we outline here:

• We compute the backreaction by assuming that the local geometry around any point on the reheating surface is RD-FRW. However, each patch will not necessarily be comoving with respect to other patches and so a global description as RD-FRW is impossible.

• We do not account for further evolution of the backreaction effect, i.e. self interaction and gravitation of the radiation fluid and its subsequent effect on the geometry. We expect that these will have a negligible effect on the galaxy motion we are interested in here.

• We ignore self-interaction of the matter fluid/galaxies and instead compute freestreaming geodesics in the modified geometry.

### 2.1 Review of geometries

In this section we will give a brief review of the geometries and establish our notation conventions, for more details as well as a full description of the causal structure the reader is directed to [15, 17].

A single Coleman-de Luccia bubble in four dimensions has an invariance, inherited from the of the Euclidean instanton [4]. When another bubble collides with ours a preferred direction is picked out, breaking the symmetry down to . As a result one can choose coordinates in such a way that the metric takes the form of a warped product between two-dimensional hyperboloids () and a 2D space.

Under the assumption that all the energy in the spacetime remains confined to thin shells or is in the form of vacuum energy, one can find the solution describing the collision by patching together general solutions to Einstein’s equations in vacuum plus cosmological constant with this symmetry [15, 14, 30, 31]. The overall result is that the spacetime inside our bubble is divided into two regions–one that is outside the future lightcone of the collision and can be described using an invariant metric (a standard open FRW cosmology), and one which is inside, is affected by backreaction, and is most conveniently described in the general case using 2D hyperboloids. However, we will find that at the late times important in our analysis, it becomes possible to once again use 3D hyperboloids and in our approximation 3D flat space (that is the geometry will be approximately locally flat FRW).

de Sitter space: The general invariant spacetime with a positive cosmological constant is given by [15]

 ds2=−dt2g(t)+g(t)dx2+t2dH22, (2.1)

where

 g(t)=1+t2ℓ2−t0t, (2.2)

and

 dH22=dρ2+sinh2ρdφ2 (2.3)

is the metric on the unit and we identify . We are most interested in observers far from the domain wall, as these see cosmologies most like ours. In that case it was shown in [17] that for the collision to be within our past lightcone we require at cosmologically relevant times. Therefore, throughout the rest of the paper we approximate giving .4 This is a metric for pure dS space (albeit in unconventional coordinates) with radius of curvature . Hereafter, we set ; it can easily be restored with dimensional analysis. These coordinates, which we denote the coordinates, are related to the or invariant ones by

 coshτ = √1+t2cos(x−x0), sinhτsinhξcosθ = √1+t2sin(x−x0), (2.4) sinhτcoshξ = tcoshρ, sinhτsinhξsinθ = tsinhρ, φ = φ, (2.5)

where the free parameter is used to fix the location of the origin of . In these coordinates the metric becomes

 ds2dS = −dτ2+sinh2τdH23,with (2.6) dH23 = dξ2+sinh2ξ(dθ2+sin2θdφ2).

It was also shown in [17] that an observer can only see an exponentially small part of the reheating surface in , that is the an observer has access to is very small (we will make this more precise shortly). Thus, if we choose appropriately we have for any observer. Throughout we make this approximation and we thus find

 dH23 = dξ2+sinh2ξ(dθ2+sin2θdφ2)≈dξ2+ξ2(dθ2+sin2θdφ2)=ds2R3. (2.7)

Radiation domination: We will also make use of the RD-FRW metric given by

 ds2RD = −dτ2+a2(τ)dH23≈−dτ2+a2(τ)ds2R3, (2.8) a(τ) = √C(τ+12), (2.9)

with a constant.5

At reheating we need to match both the scale factor and Hubble constant of the two metrics [17]. This gives

 τRDr = 12(tanhτdSr−1), (2.10) C = 2coshτdSrsinhτdSr=2a2r, (2.11)

where is the scale factor at reheating. Note that (2.10) is a gluing relation joining the two spacetimes; it should not be interpreted as a coordinate transformation. We also note that we are able to use this solution only because for the cosmologically relevant times. If this were not the case, the solution would be more complicated. Due to the reduced symmetry, the general solution for RD with invariance is not known.6

Parameters and expansions: It will be useful here to define our various parameters and orders of expansion. The number of efolds of inflation is . After reheating the temperature redshifts as so we can define

 eN∗≡Tr/T0n=an/ar, (2.12)

where the subscript refers to the observer’s time and this defines . In [17] it was shown that an observer can see a part of the reheating surface approximately in size. As is one curvature radius of the , is roughly the number of efolds required to solve the flatness and horizon problems. To agree with current observations we require and . This justifies the previous assumption that . In our computations we frequently drop subleading terms that are suppressed by or , and expand all quantities as power series in

 ϵ≡eN∗−N≪1. (2.13)

We will work to first subleading order in throughout this paper.

## 3 Reheating and matter flow

In this section we review and discuss the geometry of the reheating surface, and its effect on the post-reheating spacetime. A full treatment of the reheating surface is given in [17] and for details the reader is directed there. Here we will present a brief review. The basic idea is as discussed in Sec. 2: the collision and domain wall modify the reheating surface and break the isotropy to . We define the reheating surface as a surface of constant energy density for the inflaton field . During inflation these surfaces are also surfaces of constant value for the inflation field. This agrees with the usual slow-roll conditions so long as we are far enough away from the domain wall. Near the wall, the inflaton develops large spatial gradients and the usual slow-roll treatment is not enough. In this paper, we will focus on the region where is enough to define the reheating surface.7

### 3.1 The scalar field and reheating surface

In [17] an approximate solution for the scalar field and reheating surface was presented.8 This computation involves several steps. We need to solve the scalar field equation with a linear potential, , which is a good approximation during slow roll inflation.

The remaining inputs are the boundary conditions. We expect the scalar field to have a set of values in the other bubble, some other set of values in our bubble and a third set in the metastable vacuum outside both bubbles . In the thin wall limit the scalar field jumps from near some other local minimum (either the false vacuum our bubble tunneled from, or the vacuum inside the bubble it collided with) to a location a bit up the “hill” from our local minimum. This transition occurs all along the domain and bubble walls, so the scalar field is constant (though not necessarily the same constant) along each of these surfaces. As in [17] we choose without loss of generality along the bubble wall and along the domain wall. Note that is a possible boundary condition. The remaining condition on is that it is continuous across the radiation shell of the collision. At linear order this is enough to give us the solution for and the reheating surface as a function of , the potential and the details of the collision (such as collision time, acceleration of the domain wall, etc.). Once again for a full treatment the reader is directed to [17], where the full solution for is given.

For reasons stated above, we are interested in the reheating surface near the radiation shock, as illustrated in Fig. 1. The null line for the radiation shock was found in [17] as

where is the time of the collision, and we approximated , as reheating happens after . Expanding the solution for the inflaton around we find

where corresponds to the solution in the unperturbed case, without a collision. The step function ensures the solution is given by in the part of spacetime causally disconnected from the collision. The coefficient of the linear term is given by

 A≡−12⎛⎜ ⎜ ⎜ ⎜⎝tc−1tc−43kμ+ln[4tc1+t2c1π−2arctantc]π−2arctantc⎞⎟ ⎟ ⎟ ⎟⎠, (3.3)

up to corrections of order . For realistic values of the parameters , the slope is of the order . In the rest of this paper we will be working to first order in the parameter .9

It is useful to convert this to coordinates for use in the rest of the paper. Using the coordinate transformation (2.4) with the origin of fixed to the radiation shock by setting , we find

 (3.4)

where and we can drop the term without loss of generality as it amounts to a constant shift of and has no effect on the reheating surface. Hence in these coordinates the reheating surface consists of two flat 3D-surfaces meeting at an angle, as indicated in Fig. 2.

The form of the scalar field in (3.4) suggests that it will be convenient to switch to a cartesian system

 {ξ=√x2+y2,θ=arctanyx,⇔{x=ξcosθ,y=ξsinθ, (3.5)

so that , and thus the natural equal time surfaces, only depend on and .10

### 3.2 Matter flow and causal regions

The spacetime after reheating is separated into three causal regions, denoted I, II and III in Figs. 1 and 2. Observers in region I are causally disconnected from the bubble collision, and the spacetime in region I is RD-FRW space with time coordinate , as given in (2.8). Since region II is casually disconnected from region I, observers in region II only see the perturbed part of the reheating surface. The metric in region II is sourced by the perturbed part of reheating surface only. The symmetry of the reheating surface then implies that region II is also given by RD-FRW space, but with a time coordinate that is constant on this part of the surface. One way to view this region is as a RD-FRW space that has been boosted to be comoving with respect to the sloped reheating surface.

Our first goal will be to seek a relation between the II and I coordinates to find a global description of these regions (in fact, we will find these relations closely resemble a Lorentz boost). In region III, the metric is unknown. A full treatment would backreact the interacting radiation fluid in this region, which is beyond the scope of the current paper. Instead, we will attempt to approximate the backreacted metric in region III by taking a well motivated interpolation between regions I and II. More details on this interpolation will be given in Sec. 3.3.1

We take the matter fluid to be created at reheating, and moving orthogonal to the reheating surface. Hence, in regions I and II the matter fluid moves along geodesics defined by and respectively. This assumption is justified as any peculiar velocity given to the matter field at reheating will quickly redshift away in RD-FRW space, leaving the motion along and in regions I and II to a good approximation.

### 3.3 Relations between the regions and metrics

Region II: From the form of the scalar field (3.4), and the gluing relation (2.10), we find that the reheating surface is given in RD coordinates as

 Reheating surface: {τ=0,for% x≥0,andτ+Ax=0,for x<0. (3.6)

To define the coordinates in region II we note that on the sloped part of the reheating surface (3.6), which together with the space being RD-FRW implies

 τII=γ(τI+AxI) (3.7)

for some constant . In a coordinate basis we have , where denotes the inner product with respect to the metric, and thus

 ∂xII=~γ(A∂τI−∂xI) (3.8)

for some . We can fix the constants of proportionality at the intersection of the two parts of the reheating surface, as the metric there is given by RD-FRW in both coordinates, and we have

This gives the coordinate relations

 τII = γ(τI+AxI), xII = −γ(Aa2rτI+xI), (3.10) yII = yI,φII=φI. (3.11)

We can write the metric in region II in I-coordinates as

 dsII = −dτ2II+a2(dx2II+dy2II+y2IIdφ2II) (3.12) ≈ −dτ2I+a2(dx2I+dy2I+y2Idφ2I)+2A(a2a2r−1)dτIdxI, (3.13)

where we have dropped subleading terms suppressed by and .

Boundaries of the regions: The boundaries of regions I, II and III are defined by null surfaces originating from the surface , propagating in the -plane. This implies with , from which one can solve

 τI=(arxI+1)2−12≈a2rx2I2 (3.14)

as the equation for the boundary between regions I and III; the approximation used was . Since spacetime in region II is also RD-FRW with time coordinate , the boundary between regions II and III is given by .

#### Region III

We now define the coordinates and the metric in region III. We do this by first constructing a physically motivated ansatz for a coordinate system and interpolated metric. Then we vary this metric and interpolation, and compute certain coordinate invariant quantities to show that our ansatz produces the physically most reasonable interpolation in region III.

We first fix the direction by using the symmetry of the spacetime by setting

 ∂τIII∝∂τI+∂τII∝∂τI−A2a2r∂xI. (3.15)

By orthogonality this fixes the coordinate to be of the form

 dxIII∝A2a2rdτI+dxI,⇒xIII=h(A2a2rτI+xI), (3.16)

for some function .

Since regions I and II are causally disconnected, identical RD-FRW spaces (with a relative boost between them), we expect evolution in each region to be identical. That is, we expect an event (e.g. recombination) that occurs in region I at will occur in region II at . We are therefore motivated to take an ansatz for region III that matches from a point on the boundary between regions I and III to a point on the boundary regions II and III along a surface of constant time (), see Fig. 2. Naively, inside region III this surface could take any shape, though clearly there are choices that are more physically motivated. The simplest physically motivated ansatz we can take is to demand the surfaces of equal time are linear in (and () coordinates. We will first make this choice, and then work out a more general ansatz in Sec. 3.3.2. We will show that to the order we are working in, all of these choices give similar results.

The slope from to is independent of and this fixes

 τIII=f(τI+A2xI), (3.17)

for some function . Finally, we can fix the functions and by choosing (without loss of generality) that the coordinates are continuous across the border so that points and are given in III-coordinates as and .11 With these constraints the inverses of the functions are fixed as

 f−1(s) = s+A2ar√2s, (3.18) h−1(s) = s+sign(s)A4s2, (3.19)

where is a dummy variable, and the ensures that is an odd function in . These can be inverted to find and , but for our purposes these inverses are sufficient.

Note that the coordinate system thus defined respects the symmetry of the spacetime; in II-coordinates we have

 τIII = f(τII+A2xII), (3.20) xIII = −h(A2a2rτII+xII). (3.21)

Using the coordinate relations above and the RD-FRW metric (2.8), we can find the metric in regions I and II in III-coordinates. We compute these metrics at the boundaries12 of region III, and then define the metric in region III as a linear interpolation in on slices of constant between the boundaries.

Using the relations (3.16) and (3.17), and the metric (2.8), we can write the metric at point as

 ds2III|P=−dτ2III+a2(1+Aar√2τ0)dx2III+a2(dy2III+y2IIIdφ2III)−Aa2a2rdτIIIdxIII, (3.22)

where here , and we have again neglected subdominant terms suppressed by , and . Similarly, we can find the metric at the opposite point

 ds2III|~P=−dτ2III+a2(1+Aar√2τ0)dx2III+a2(dy2III+y2IIIdφ2III)+Aa2a2rdτIIIdxIII. (3.23)

Since and were arbitrary points on the boundaries, these relations specify the metric everywhere on the boundaries of region III.

The interpolated metric: We define the metric in region III as a linear interpolation in on the constant timeslice between (3.22) and (3.23). Since 13, the linear interpolation on the timeslice between and gives

 gIIIττ(τ0,xIII)≡gIIIττ(P)=−1. (3.24)

Likewise, and have the same value at and , and hence we fix those components to be independent of on the timeslice .

The remaining component is , which changes sign between points and . The linear interpolation then defines on the timeslice as

 gIIIτx(τ0,xIII)≡gIIIτx(P)⋅(xIII√2τ0/ar)=−AaxIII2, (3.25)

which reduces to the correct boundary values as .

We have completely defined the interpolated metric on the timeslice , and since was arbitrary, this defines the metric in all of region III as

 ds2III=−dτ2III+a2(1+Aar√2τIII)dx2III+a2(dy2III+y2IIIdφ2III)−AaxIIIdτIIIdxIII. (3.26)

#### A general coordinate system

The previous ansatz was the simplest physically motivated one. However, one could argue that a more general interpolation could capture the physics in region III better. In this section we will find a more general coordinate system, constrain it by physical considerations and then show that the effect on the galaxy flow is minimal between these choices. We vary our coordinate system by defining

 {T(τIII,xIII)=τIII+Aδ1(τIII,xIII),X(τIII,xIII)=xIII+Aδ2(τIII,xIII), (3.27)

where and are a priori arbitrary, but will be constrained by physical considerations. The factors of have been added for later convenience. Without loss of generality we pick and to vanish on the boundaries of region III, to keep the coordinates continuous across the boundary. With this choice, it can be shown that the magnitudes of the s are and . If and were bigger, and would not be timelike and spacelike respectively for generic and .14

Using (3.27) and (3.16,3.17) we solve for and , and write the metrics in regions I and II in -coordinates. As above, we then interpolate linearly in across region III, giving the metric in region III as

 ds2III=−dT2+a2(T)(1+Aar√2T(1−δX2))dX2+a2(dy2III+y2IIIdφ2III)−AaX(1−δT2)dTdX, (3.28)

with , and we have defined

 ⎧⎨⎩δT2≡−sign(X)2a2r∂Tδ2(T,X),δX2≡2ar√2T∂Xδ2(T,X). (3.29)

The appearance of above can be traced back to the the sign change across the boundary. Due to the considerations above we generically have . Note that does not appear in the metric to this order, and thus we are free to set it to vanish, giving .

We must demand that the boundaries between regions I and III (and II and III) are null in this metric. The boundary is generated by the vector , and requiring this vector to be null gives

 δT2(T,X)|b=δX2(T,X)|b, (3.30)

where the indicates that this relation is valid on the boundary.

Curvature in region III: At this stage, we have some constraints on the s, but a lot of freedom remains. We can ask what physical expectations we have about the behavior of physical quantities in region III and use that to further constrain the functions. For example, if the s were wildly varying functions, then the curvature in region III would be large in places, varying on small scales and possibly even negative. This behavior is clearly unphysical. We are thus motivated to examine the curvature scalar along a constant time surface to aid in constraining the s. We will find that with this criteria as well as symmetry considerations there are two natural choices, and the difference in flow velocities between the two choices is at most .

The curvature scalar in region III is

 R=2Aar(2T)32(3+3a2r∂Tδ2+ar(3arX−4√2T)∂T∂Xδ2+2Ta2r∂2Tδ2+2Tar(arX−√2T)∂2T∂Xδ2). (3.31)

The constraints on to a large degree fix its functional form. By continuity of the coordinates must vanish as , and by the symmetry of the spacetime it must vanish as . Furthermore, as we have , a fairly general15 polynomial function we can write down that satisfies (3.30) and the boundary conditions is

 δ2(T,X)=F(arX√2T)arX√2T(X2−2Ta2r), (3.32)

where is an unknown function of at most order unity, and .

At the boundary between regions I and III we have , and hence we expand for some constants . We are mostly interested in geodesics relatively close to the boundary, so to a good approximation we can truncate

 F(arX√2T)≡b0+b1(1−arX√2T). (3.33)

While this is clearly not the most general form of in the interior of region III, it captures the physics close to the boundary, and by a suitable choice of and we ensure that physics in the interior is also reasonable. We fix the constants and by demanding that the curvature scalar vanishes on the boundary of the regions, and that the curvature scalar is differentiable at , as there are no sources along . Plugging (3.32) into (3.31) we find that these constraints imply

 b0=−b1=−310. (3.34)

Note that the choice gives the coordinate system and corresponds to a constant curvature across region III proportional to . In Fig. 3 we plot the scalar curvature on a constant time slice across the spacetime, for both and .

In the next section we will show that the matter flow is largely unaffected between these two cases; the flow velocities in the are roughly 20%-30% higher than in the case. Thus our conclusions are fairly robust under the choice of the interpolated metric.

## 4 Galaxy motion

In the previous section we found a physically motivated metric for region III. Together with the metrics in regions I and II and the various coordinate relations we have a global description of the spacetime background to order . In this section, we will find the motion of galaxies through this spacetime and show that the collision generically leads to peculiar velocities in the affected region.

In our approximation we will assume the matter that makes up galaxies was created at reheating, and was initially moving orthogonal to the reheating surface, i.e. initially generated by and in regions I and II respectively. We track the future evolution of these geodesics and analyze the motion of the galaxies as seen by present day observers.

### 4.1 Timelike geodesics in region III

We are interested in observers originating in region I, who have entered region III in relatively recent times16. We define the time of the observer to be . This implies the observer is at ; for larger values of the observers will not yet see the collision. We will find the timelike geodesics in region III for this range of parameters.

Galaxies that originate in region I follow the geodesics with constant and while in region I. Upon entering region III they have four-velocity

 u0 = ∂τI=uτ0∂τIII+ux0∂xIII,with (4.1) uτ0 = 1−O(ϵ2),ux0=A2a2r(1+O(ϵ2)), (4.2)

where and , and .17 18

We are thus interested in timelike geodesics in region III with initial velocity given above. Since the velocities are non-relativistic19, the normalization condition gives

 uτ=1−O(ϵ2),⇔τ=s(1+O(ϵ2)), (4.3)

where is the affine parameter on the geodesic. Making use of (4.3), we find the geodesic equation for to first order in to be

 d2xIIIdτ2III+1τIIIdxIIIdτIII−AxIII2ar(2τIII)32=0. (4.4)

This can be solved in the relevant range of coordinates to give

 xgeod.III(τIII)=x0+Aτ0a2r[(√τIIIτ0−1)−12(1−a2rux02A)lnτIIIτ0]+O(ϵ3), (4.5)

where is the point where the geodesic intersects the boundary between regions I and III, as shown in figure 5, and is the -component of the four-velocity at that point.20

Peculiar velocity: Using the initial velocity found in (4.1) and the solution for the geodesic (4.5), we find that at time , the velocity of a galaxy which entered region III at time is given by

 vIIIgal=dxgeod.IIIdτIII=A2a2r√τ0τIII. (4.6)

Note that by velocity we mean velocity in coordinate system III. To convert these to physical velocities, one must take into account the velocity due to the expansion of the universe, i.e. Hubble’s law and convert to the CMB rest frame.

The general coordinate system: The procedure above can be repeated for the -coordinate system. In this case, the geodesic equation becomes

 d2XdT2+1TdXdT−AX2ar(2T)32⎡⎣1+4(b0+b1)(arX√2T)3−12b1(arX√2T)4⎤⎦=0, (4.7)

and the initial velocity when the galaxy enters region III in these coordinates becomes

 uX0=A2a2r(1−4b0). (4.8)

The geodesic equation can be solved perturbatively to first non-trivial order in to give

 Xgeod.(T)=X0+AT0a2r⎡⎣(√TT0−1)−(b0−b13)+(b0+b1)(T0T+lnT0T)−43b1(T0T)32⎤⎦, (4.9)

which reduces to (4.5) as . Here is the point where the galaxy enters region III. Differentiating, we find the velocity along the geodesic as

 v(T,X)gal=dXgeod.dT=A2a2r[√T0T−2(b0+b1)(T0T+T20T2)+4b1T40T4]. (4.10)

To assess the effect of the choice of the interpolating coordinate system on the matter flow, we convert the velocity above to the (approximate) CMB rest frame21 which is given by the coordinates of region I using the coordinate transformations (3.27), (3.17) and (3.16). This gives

 vrestgal=−A2a2r(1−√τ0τI)⎛⎝1−2(b0+b1)⎡⎣√τ0τI+(τ0τI)32⎤⎦+4b1(τ0τI)2⎞⎠, (4.11)

and in Fig. 4 we plot these velocities for the two preferred sets of values, and . Both sets lead to comparable flow velocities, with the case generally leading to velocities that are roughly 20-30% higher than the case. This shows that our model is fairly robust under the choice of a (physical) interpolating metric. For this reason, and for computational simplicity, we focus on the case from now on; the results obtained for the other choice of interpolation will be similar, but would require numerical analysis.

### 4.2 Past lightcone of observers in region III

In the previous section we found a general expression for the peculiar velocity of galaxies relative to the CMB rest frame (4.11). We would like to find the observable consequences of these peculiar velocities, i.e. how big a patch of the sky they take up at a given redshift and how large the velocities are for reasonable scenarios. In this section we will work out how the size of the disk on the sky that contains galaxies with non-vanishing peculiar velocity varies as a function of , , the redshift to the galaxies in question and also in terms of the effect on the CMB sky from the collision found in [17]. To do this, we take an observer who has recently entered region III, project back his past lightcone to a given redshift and then find the galaxies behavior within this lightcone. We will focus on redshifts of approximately .

Consider observers at point in region III (and region III coordinate values), travelling along one of the geodesics found above (4.5). We wish to analyze the motion of galaxies as seen by these observers. As neither the metric nor the scalar field depend on up to order , we are free to choose the origin of the plane to coincide with the location of the observers, so that . To find the past light cone, we define a coordinate system centered on the observers, , related to the usual region III coordinates by

 τobs=τIII, xobs=xIII−xn, (4.12) yobs=yIII, φobs=φIII. (4.13)

We can find the null geodesics projected back from the observer in a similar manner to the procedure for finding the timelike geodesics of the previous sections

 τobs(ξobs)=12(


text:
# Math Help - Present Annuity?

1. ## Present Annuity?

John wishes to set up an account for his grandfather so that he can have some extra money each month. John wants his grandfather to be able to withdraw $110.00 per month for the next 2 years. How much must John invest today at 7% per year compounded monthly so that his grandfather can withdraw$110.00 per month for the next 2 years?

I am unsure of the formula to use for this problem. Is this a present annuity problem? If so, how do I work it?

2. I read this problem in two possible ways. If John wants to make this investment before his grandfather starts withdrawing is one scenario and another is if his grandfather will start withdrawing the same month as the investment. Which is it?

I am guessing the first and if so, first consider how much money \$110 a month for two years is in total. Then you can use a basic compound interest formula to solve for the principal amount needed.


text:
Locally compact, complete, bounded spaces - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-21T09:06:24Z http://mathoverflow.net/feeds/question/73170 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73170/locally-compact-complete-bounded-spaces Locally compact, complete, bounded spaces Pablo Pigeon 2011-08-18T17:40:06Z 2011-08-18T20:13:34Z
Is every locally compact, complete, bounded space compact?

It seems that this fact is implicitly used in the definition of Gromov-Hausdorff convergence for pointed spaces on wikipedia, but I'm not able to prove it.
 http://mathoverflow.net/questions/73170/locally-compact-complete-bounded-spaces/73173#73173 Answer by Stefan Geschke for Locally compact, complete, bounded spaces Stefan Geschke 2011-08-18T18:47:46Z 2011-08-18T20:13:34Z
A metric space is compact if and only if it is complete and totally bounded. This is the Heine-Borel theorem. Is this what you are looking for?
 http://mathoverflow.net/questions/73170/locally-compact-complete-bounded-spaces/73174#73174 Answer by Mahdi Majidi-Zolbanin for Locally compact, complete, bounded spaces Mahdi Majidi-Zolbanin 2011-08-18T18:48:50Z 2011-08-18T18:48:50Z
I don't believe this is true. For example, take an infinite set and put discrete metric on it, that is, \$d(x,y)=0\$ if \$x=y\$ and \$d(x,y)=1\$ if \$x\not=y\$. Then I believe this is locally compact, complete and bounded, but it is not compact. Did I understand your question correctly?