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Concerning Quality

# Quality and Paradigm: The Assembly Language of Reasoning and the Domain-Specific Language of the Machine

Does programming paradigm affect the ultimate quality of software? I generally avoid stoking the eternal flame war of functional vs. imperative programming, but it has to be discussed since there are so many repeated statements of fact about which one leads to higher-quality programs1. The arguments primarily exist along two dimensions: correctness and performance. Summing them up, functional programs are more correct but slow, and imperative programs are buggy but fast. Of course, there are counterexamples to each of these statements, but this is a fair generalization overall.

Thinking of it as a competition isn’t helpful though, because each paradigm has a different purpose: FP is optimized for mathematical reasoning, whereas imperative programming is fundamentally about manipulating computer hardware. It’s this difference between reasoning and practical execution that suggests that the highest quality software actually requires utlizing both paradigms: no matter what level we write code at, we must be able to reason about how that code runs at the level of real-world machines. This is what allows us to achieve the holy grail of quality: fast and correct software.

Destructive array updates are a great backdrop for analyzing the differences between the FP and imperative styles. We’ll use C for the imperative example:

In the imperative style, updating the value at index 1 in a mutates a, meaning printing a afterwards would show [1, 4, 3]. The 2 is gone, making the update destructive. This is a notoriously difficult example for FP to handle, which leads to its detractors to (justifiably) criticize it. How can a paradigm be considered serious if it can’t handle one of the most common programming techniques there is?

This begs two questions:

1. What exactly is difficult about modeling an array in FP?
2. Why is it so convenient in an imperative language?

To look into #1, here’s a first attempt at modeling the above imperative code in an FP style, in Isabelle/HOL2:

fun update :: "'a list ⇒ nat ⇒ 'a ⇒ 'a list" where
"update [] i v = []" |
"update (x # xs) i v =
(case i of 0 ⇒ v # xs | Suc j ⇒ x # update xs j v)"

definition "fp_list = [1, 2, 3::int]"

value "update fp_list 1 4" (* Returns [1, 4, 3] *)
value "fp_list" (* Returns [1, 2, 3] *)

We use a list to model an array since we can represent a list as a recursive datatype directly, e.g. (Cons 1 (Cons 2 (Cons 3 Nil))). We have an update function which uses recursion and pattern matching on the list values to build up the updated value from scratch, with update fp_list 1 4 equaling the desired value: [1, 4, 3]. However, this function does not have the semantics of the array update because fp_list's value remains unchanged after being passed as an argument to update. The value returned by update is not connected to or tied to the value of fp_list in any way, and this is because that is simply the semantics of mathematical functions - they only map input values to output values, i.e. they are pure.

To achieve destructive update semantics, we need to add the notion of a “reference.” A reference is an indirection, a way to refer to a value by something like a name or an address. There are a million ways to do this, but here’s a simple approach:

type_synonym var_name = "string"
type_synonym state = "var_name ⇒ int list"

definition read_state :: "state ⇒ var_name ⇒ int list" where
"read_state s var = s var"

definition write_state :: "state ⇒ var_name ⇒ int list ⇒ state" where
"write_state s var l = s(var := l)"

definition imp_update :: "state ⇒ var_name ⇒ nat ⇒ int ⇒ state" where
"imp_update state var n idx =
(let list = read_state state var in
write_state state var (update list n idx))"

definition "state ≡ λ v. (if v = ''a'' then fp_list else [])"

value "read_state (imp_update state ''a'' 1 4) ''a''" (* [1, 4, 3] *)

Our new function, imp_update, does have the destructive semantics we’re looking for, but in order to get it we introduce the ability to reference a value by name. We do this by modeling state as a function from variable names to values (with the only values we can have being lists of integers to keep things simple). Then, instead of operating on values directly, we must read from and write to this state by specifying the referred variable’s name - in this case a. It’s worth calling out that in write_state, the syntax s(var := l) is actually creating a new function that has all the same mappings as s except with var now mapped to l. Isabelle/HOL allows us to manipulate functions in the true mathematical sense where they are just sets (of ordered pairs), and in that way they are the reasoning analog of hash tables.

The final evaluation shows that when we update a with imp_update and we read its value back out of the state, a does have the modified value of [1, 4, 3].

This answers our first question: destructive updates are such a pain to model in an FP language because FP only has the notion of mathematical functions which are pure and don’t modify their arguments. It can be done, but it requires manually modeling state and variable references. This leads to all variable accesses being polluted with an extra step - the state must be consulted to get anything done.

It’s this point that immediately leads to the answer to our second question: destructive updates are so natural in imperative languages because in them, state and references are implicit.

To dive into that, here’s our initial C code again:

and here’s is the assembly code (ARM) that results from compiling3 it:

	.section	__TEXT,__text,regular,pure_instructions
.build_version macos, 11, 0
.globl	_main                           ; -- Begin function main
.p2align	2
_main:                                  ; @main
.cfi_startproc
; %bb.0:
sub	sp, sp, #64                     ; =64
stp	x29, x30, [sp, #48]             ; 16-byte Folded Spill
add	x29, sp, #48                    ; =48
.cfi_def_cfa w29, 16
.cfi_offset w30, -8
.cfi_offset w29, -16
ldr	x8, [x8, ___stack_chk_guard@GOTPAGEOFF]
ldr	x8, [x8]
stur	x8, [x29, #-8]
str	wzr, [sp, #12]
ldr	q0, [x8]
str	q0, [sp, #16]
mov	w9, #4                   ; Move the value 4 into a register
str	w9, [sp, #24]            ; Store that value at memory location a[1]
ldr	x8, [x8, ___stack_chk_guard@GOTPAGEOFF]
ldr	x8, [x8]
ldur	x10, [x29, #-8]
subs	x8, x8, x10
b.ne	LBB0_2
; %bb.1:
mov	w8, #0
mov	x0, x8
ldp	x29, x30, [sp, #48]             ; 16-byte Folded Reload
add	sp, sp, #64                     ; =64
ret
LBB0_2:
bl	___stack_chk_fail
.cfi_endproc
; -- End function
.section	__TEXT,__literal16,16byte_literals
.p2align	2                               ; @__const.main.a
l___const.main.a:
.long	1                               ; 0x1
.long	2                               ; 0x2
.long	3                               ; 0x3
.long	4                               ; 0x4

.subsections_via_symbols

This has all of the compiled assembly for completeness, but highlighted is a mov instruction followed by a str instruction. Recall that a “move” instruction (mov) places a value into a register, and a “store” instruction (str) takes a value in a register and writes it to a location in memory. These two instructions are what the statement a[1] = 4 get compiled into, and hopefully at this point a light bulb is turning on: imperative programs assume an underlying CPU and memory. The instructions might be slightly different depending on the processor, but destructive updates are simply taking advantage of memory instructions that every modern CPU supports.

Even though functional languages have to worry about executability at some point, the paradigm of FP exists without any assumption of an underlying machine since its semantics is simply the semantics of math. This brings us to the difference between reasoning and execution.

# Reasoning vs. Execution

While the line between them can seem blurry, reasoning and execution are different activities. Reasoning is logical thinking. Instead of blindly accepting a conclusion, we use reasoning to justify the steps of thought that were taken to reach it. But reasoning is abstract, and there is no difference in cost to your mind between reasoning about tiny fractions or gargantuan numbers. Even infinite subjects, like the set of all even numbers greater than 9,000, can still be reasoned about - we know that all numbers in this set are greater than 5, for example.

In contrast, execution is about running specific programs on specific machines. This is where implementation details such as instruction set architecture and word sizes have to be considered, and also where time and space really matter. A program might even have an infinite loop, as all interactive programs do, but we will still only practicaly execute it a finite number of times because we want to know the result of the program in our lifetime. For this reason, along with the practical limitations of physical hardware, execution is inherently finite, and we can only execute a subset of what we can reason about. This distinction is notable, because each programming paradigm is optimized for one activity at the expense of the other.

We often say that functional programming is higher-level than imperative, with imperative being lower-level and closer to the machine. But this is only true when considering execution. FP is “closer to the metal” of pure reasoning, whereas imperative programming is higher-level reasoning. This is because, again, functional programming is based on the semantics of math, which is foundational and can be seen as the “assembly language” of reasoning. With respect to reasoning, imperative programming is more like a domain-specific language that can be “compiled” to math’s semantics by explicitly modeling computer memory with state and variable references in mathematical terms (as we saw in our previous example).

Destructive array updates are the perfect example of this. The imperative code for our example is way more compact because it’s written using an effective DSL. The CPU and memory state are abstracted away from the syntax, though they are still part of the semantics, i.e. what the program actually does. This is the hallmark of DSLs - they make expressing concepts in a particular domain more natural by leaving parts of it implicit.

So FP and imperative programming mainly differ in their acknowledgement of computer hardware and their adherence to mathematical semantics. FP is programming with reasoning, and imperative programming is programming with a machine. They intersect at the aforementioned holy grail of formally verified, fast executable programs. For that, we need to be able to reason about imperative programs, i.e. reason about execution on a real computer.

Before software can be formally reasoned about, it must first be represented in some form of logic.4

We saw the difficulty with destructive array updates in a functional language. Since FP is closer to pure reasoning, this dificulty directly translates to reasoning about code in a compiled imperative language such as C. There’s tons of research about how to best reason about imperative programs, but I’ll highlight one approach here: translating code into a logic as a shallow embedding and reasoning via Hoare triples. In this case, the logic is higher-order logic - the flavor that Isabelle/HOL implements. Once we’re in the logic, we can determine the truth or falsity of statements with pure reasoning.

Let’s slightly modify our example so that it operates on arbitrary arrays instead of the one hardcoded one:

The behavior of update_arr is very simple: it is correct if the passed in array is updated at index 1 with the value v.

Let’s pull it into Isabelle for reasoning:

theory ImperativeReasoning

imports AutoCorres.AutoCorres

begin

install_C_file "array_update.c"
autocorres[heap_abs_syntax] "array_update.c"

context test begin

definition "array_mem_valid s a = (is_valid_w32 s a ∧ is_valid_w32 s (ptr_add a 1))"

theorem
"validNF
(λs. array_mem_valid s a ∧ s = s0)
(update_arr' a v)
(λ_ s. (array_mem_valid s a) ∧ s = s0[(ptr_add a 1) := v])"
unfolding update_arr'_def and array_mem_valid_def
apply(wp)
apply(auto simp: fun_upd_def)
done

There’s actually quite a bit going on in this short example, so let’s unpack it. First, this is using AutoCorres, which is a tool for reasoning about C code via an Isabelle/HOL representation. The install_C_file and autocorres commands parse and translate the C code of our example, with autocorres performing several abstractions and optimizations to the code that greatly simplifies reasoning about it in Isabelle. For what we’re talking about, the most relevant abstraction is that the concept of a heap is added to the translated C code. The heap is a model of the underlying memory that the C program interacts with via pointers, arrays, mallocs, or any other memory-manipulating constructs. Because we are going to be reasoning about the program, the memory must be modeled explicitly, as we did in the functional destructive array update code. AutoCorres has a more advanced design of the heap, which you can read more about in David Greenaway’s PhD thesis. I recommend at least reading the chapter about heap abstraction, it’s pretty digestible and very informative.

Next, let’s unpack the theorem that we’ve constructed:

theorem
"validNF
(λs. array_mem_valid s a ∧ s = s0)
(update_arr' a v)
(λ_ s. (array_mem_valid s a) ∧ s = s0[(ptr_add a 1) := v])"

In math, a theorem is just a statement that is proven to be true, and the primary way that we can reason about anything is to come up with some such statement and try and evaluate its truth or accuracy. One of the most common ways of expressing statements about an imperative program is with Hoare logic, which is what we’re doing in this theorem. validNF p c q is a function that represents a Hoare triple, which is a specification of what a program or part of a program should do in order to be correct. It takes three arguments:

• p is a precondition. This must be true before executing the command c.
• c is a command. This is some program statement or set of statements that we’re looking to reason about.
• q is a postcondition. This must be true after c is executed.

Our precondition is: (λs. array_mem_valid s a ∧ s = s0), where we defined array_mem_valid s a as is_valid_w32 s a ∧ is_valid_w32 s (ptr_add a 1). The precondition is actually a function where the argument s is the current state of the program, which includes the current heap state. is_valid_w32 s a checks that a pointer a in state s is “valid”, where validity is pointer validity as described by the semantics of C. For example, for a pointer to be valid, it must not be NULL. The intention here is to guarantee memory safety - if for any reason the starting address of a has become NULL, this precondition is not met and update_arr would fail if it were called. We also add the condition that s = s0 which is just so that we can reference this initial state in the postcondition later on, a common pattern when using Hoare triples.

Then, our command is: update_arr' a v. This is just calling the translated update_arr function with arbitrary arrays a and values v.

Finally our postcondition is: λ_ s. (array_mem_valid s a) ∧ s = s0[(ptr_add a 1) := v]. This is a function whose first value we’ll ignore (_), and whose second value is again the program state, but after executing the command. We again check for pointer validity, ensuring that update_arr' is memory safe. The second part of the condition, s = s0[(ptr_add a 1) := v], says that the only change to the program state is that address a + 1 is now set to the v passed into update_arr'. This ensures that no other random memory updates have occurred.

The “NF” in validNF stands for “non-failure” which means that we are also saying that, as long as the precondition is met, update_arr' will execute to termination with no failures, known as “total correctness.” A more traditional way of representing Hoare triples is by writing: {P}C{Q}, and it can even be written this way in Isabelle using custom syntax rules, but we wrote it using a plain function call to show that there’s no magic going on under the hood. Here are the definitions that make up validNF:

validNF P f Q ≡ valid P f Q ∧ no_fail P f
valid P f Q ≡ ∀s. P s ⟶ (∀(r,s') ∈ fst (f s). Q r s')
no_fail P m ≡ ∀s. P s ⟶ ¬ (snd (m s))

The definitions themselves aren’t exactly important for this exampe, but I’m sharing them to show that this theorem amounts to an equation, and we ultimately want to prove that the left hand and right are equivalent.

To recap: we translated our actual C code into Isabelle/HOL, and then we wrote a Hoare triple which specifies what it means for update_arr' to be correct. If this call to validNF can be proven to be true, then update_arr' can be taken to be correct with respect to this specification. With a few proof statements, we do get Isabelle to accept this statement as proven:

unfolding update_arr'_def and array_mem_valid_def
apply(wp)
apply(auto simp: fun_upd_def)
done

AutoCorres provides the proof tactic wp which converts a Hoare triple into the weakest precondition needed to be satisfied in order for the postcondition to hold, an old trick from Djikstra that allows us to reduce a Hoare triple into a single logical formula that we need to prove. One absolutely invaluable benefit of moving from C to higher-order logic for reasoning is that we can use functional programming with a bit of added logic on top to define the semantics of a program along with Hoare triples about it. Once we do that, we can prove all kinds of things using only term simplification, which is what the unfolding update_arr'_def and array_mem_valid_def and apply(auto simp: fun_upd_def) lines mean - the proof is literally just a transformation of the code until the left hand side of an equation exactly matches the right hand side. This is only possible because of the heap being explicitly modeled, and couldn’t be done by using the imperative code directly.

# Outro

By using each paradigm for what it’s best at, we can formally reason about execution all the way down to the machine level when necessary. That isn’t to say that we should only reserve FP for reasoning within a proof assistant. There are plenty of optimizations out there that can translate FP code into efficient machine code, the classic example being tail-call elimination. But, sometimes, embracing the machine is the right call and can lead to unparalleled performance improvements. There are also times where imperative code is just inherently more clear, with nothing to do with performance. As they say, choose the right tool for the right job.

Understanding the strengths of each paradigm, though, rather than seeing either as a one-size fits all solution to all of software’s problems is the key to determining which one is right and when.

1. We’ll leave out other paradigms from this discussion, such as logic programming, acknowledging that FP and imperative aren’t the only possible programming paradigms.

2. The list_update function is then the same as the implementation in the List Isabelle/HOL theory, so this is more or less how an update is implemented in pure FP.

3. Compiled with clang -S

4. From Automated Proof-producing Abstraction of C Code, David Greenaway’s PhD’s thesis.


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# What recursive T(N) function typically can conclude the algorithm is O(n ^ 2), O(n log n), O(n), and O(log n)?

Is it true that some common forms of recursive T(n) can give the following conclusions?

When

T(n) = T(n/c) + b    where c is a constant > 1, b is any constant

then the algorithm is O(log n).

When

T(n) = T(n/c) + T(n/d) + bn   where c and d are constants > 1, b is any constant

then the algorithm is O(n log n).

When

T(n) = T(n - c) + bn   where c, b are constants > 1

then the algorithm is O(n2) and seems like many useful algorithms don't have this pattern and O(n2) is not often seen in classical algorithms.

I have seen the form:

T(n) = T(n/c) + T(n/d) + O(n)   where c and d are constants > 1

for the selection / median algorithm and it is concluded that the algorithm is O(n) but isn't the T(n) the same as formula 2 for O(n log n) above?

• Note that recurrences are not at all necessarily tied to algorithm runtimes. – Juho Aug 17 at 20:51
• Your question is answerable simply by taking the master theorem and playing around with different values. – dkaeae Aug 19 at 6:55

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# How is $[0,1]$ locally compact if 0 and 1 do not have a neighborhood in $[0,1]$?

I am a physics student and I am trying to learn the concept of compactness as I need it to understand some group theory issues. I am having trouble understanding the statement that every compact space is locally compact. I understand the open cover definition of compactness and could prove that $[0,1]$ is compact using the supremum method. Now, according to the definition on Wikipedia of local compactness, a topological space $X$ is locally compact if every point in $X$ has a compact neighborhood. My understanding is that neighborhood of a point $p$ in $X$ should contain an open set containing $p$ itself. It seems to me though that $0$ and $1$ in $[0,1]$ do not have open sets/intervals in $[0,1]$ which contain $0$ and $1$. Am I making a conceptual mistake here ? I am not an expert, so please forgive my stupidity here. Thank you.

• You must distinguish between a subset of $\Bbb R$ and a subspace of $\Bbb R.$ – DanielWainfleet Jan 11 '18 at 5:42
• For this reason, officially, on math exams in high schools in Norway, a function on a domain with end points (say $f:[0,1]\to \Bbb R$ with $f(x) = x$) is not considered to have local max and min at those end points. They may be global extreme points, if the function value there is larger or smaller than any other place in the domain. But since "$f$ isn't defined on a neighbourhood" (paraphrased), they can't be local maxima or minima. So don't worry, you're not the only one who's confused about this. (Also, it infuriates me to no end that we are teaching this misconception to students.) – Arthur Jan 11 '18 at 8:40

The open sets of $[0, 1]$ (in the topology we usually think of*) are the intersections of open sets in $\Bbb R$ with $[0, 1]$. So for instance, because $(-0.3, 0.2)$ is open in $\Bbb R$, we know that $[0, 0.2)$ is open in $[0, 1]$.

* (This is called the "subspace topology", by the way.)

Does that help?

• Thank you for your time, John. I think now I see what I was missing. – singularity Jan 11 '18 at 2:37

It's important to bear in mind the topology on $[0,1]$, which is the subspace topology inherited from $\mathbb{R}$. Conferring with the definition of the subspace topology, you'll find that $[0,a)$ is an open set in $[0,1]$ for $0<a<1$, though indeed such a set is not open in $\mathbb{R}$.

As an aside, there is a theorem regarding local compactness for which this scenario arises as a special case:

Theorem: Let $X$ be a locally compact Hausdorff space. If $U$ is either an open or a closed subset of $X$, then $U$ is itself locally compact.

Proof: We want to show that any point $x \in U$ has a compact neighborhood $L_x \subset U$. To do this, we begin by noting that the local compactness of $X$ guarantees that $x$ has a compact neighborhood $K \subset X$. This will serve as a starting point.

The intersection of a closed set and a compact set is itself compact, so our strategy going forward will be to find a (finite) collection of closed neighborhoods $\{ C_i \}$ of $x$ such that $\displaystyle K \cap \left( \bigcap_i C_i \right)$ is a proper subset of $U$ (necessarily compact). A good step in the right direction is to take $K \cap \overline{U}$, which prunes away many points in $K \setminus U$.

To get a third closed set $S$ so that $S \cap K \cap \overline{U} \subset U$, we can do the following: note that $\partial U \cap K$ is compact because $\partial U$ is closed (where $\partial U$ denotes the boundary of $U$), and since $X$ is Hausdorff, we can cover $\partial U\cap K$ with open sets that are "far" from $x$. Let $T$ be the union of the open sets contained in the finite subcover this will necessarily admit, and let $S = X \setminus T$ (which is a closed neighborhood of $x$).

$L_x = S \cap K \cap \overline{U}$ is a compact neighborhood of $x$ contained in $U$.

In this scenario, of course, $X = \mathbb{R}$ (a locally compact Hausdorff space), and $U = [0,1]$.


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# memo_has_moved_text(); Zeta functions for two-dimensional shifts of finite type

Jung-Chao Ban, Department of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan., Wen-Guei Hu, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan., Song-Sun Lin, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan. and Yin-Heng Lin, Department of Mathematics, National Central University, ChungLi 32054, Taiwan.

Publication: Memoirs of the American Mathematical Society
Publication Year 2012: Volume 221, Number 1037
ISBNs: 978-0-8218-7290-1 (print); 978-0-8218-9457-6 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00653-8
Published electronically: March 23, 2012
Keywords: Zeta functions, shift of finite type, Ising model
MSC (2010): Primary 37B50, 37B10, 37C30; Secondary 82B20, 11M41

View full volume PDF

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Chapters

• Chapter 1. Introduction
• Chapter 2. Periodic patterns
• Chapter 3. Rationality of $\zeta _n$
• Chapter 4. More symbols on larger lattice
• Chapter 5. Zeta functions presented in skew coordinates
• Chapter 6. Analyticity and meromorphic extensions of zeta functions
• Chapter 7. Equations on $\mathbb {Z}^2$ with numbers in a finite field
• Chapter 8. Square lattice Ising model with finite range interaction

### Abstract

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $\zeta ^{0}(s)$, which generalizes the Artin-Mazur zeta function, was given by Lind for $\mathbb {Z}^{2}$-action $\phi$. In this paper, the $n$th-order zeta function $\zeta _{n}$ of $\phi$ on $\mathbb {Z}_{n\times \infty }$, $n\geq 1$, is studied first. The trace operator $\mathbf {T}_{n}$, which is the transition matrix for $x$-periodic patterns with period $n$ and height $2$, is rotationally symmetric. The rotational symmetry of $\mathbf {T}_{n}$ induces the reduced trace operator $\tau _{n}$ and $\zeta _{n}=(\det (I-s^{n}\tau _{n}))^{-1}$. The zeta function $\zeta =\underset {n=1}{\overset {\infty }{\prod }} (\det (I-s^{n}\tau _{n}))^{-1}$ in the $x$-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the $y$-direction and in the coordinates of any unimodular transformation in $GL_{2}(\mathbb {Z})$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $\zeta ^{0}(s)$. The natural boundary of zeta functions is studied. The Taylor series for these zeta functions at the origin are equal with integer coefficients, yielding a family of identities, which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions.


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# Is the black brane picture at strong string coupling?

+ 2 like - 0 dislike
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As far as I am aware there are two pictures of branes:

• for $g_SN\gg1$: branes as solitonic solutions to the field equations of supergravity (i.e. black branes)
• for $g_SN\ll1$: branes as fundamental objects, on which we can build $\sigma$-models (e.g. the fundamental string).

where $g_S$ is the string coupling and $N$ is the units of flux or number of branes in a stack.

My understanding is that these two are seen as complementary pictures of the same objects, e.g. a condensate of fundamental $p$-branes turns into a black brane. However, I am not sure whether the black brane picture is valid at strong string coupling too. Some seemingly incompatible excerpts from the literature:

• In Maldacena's 1997 paper, it is mentioned that the black brane limit $g_SN\gg1$ is understood to be at small $g_S$. Indeed, in the classical review hep-th/9905111, the limit is written as $N>g_SN\gg1$ to emphasise this.
• From Clifford Johnson's D-brane primer: $g_SN\gg1$ "corresponds to either having $N$ small and $g_S$ large, or vice versa".

So I am not sure what to believe: are black branes (as solutions to supergravity) at small $g_S$, or at large $g_SN$ with $g_S$ free to take any value (as long as $g_SN\gg1$)? I see reasons for both:

• Since we are talking about (10-d) supergravity solutions, we ought to be at small $g_S$ (in addition to small $\alpha'$), since SUGRA is the low-energy limit of string theory (which is perturbative in $g_S$). However, I guess as we increase $g_S$ we would simply go to 11-d SUGRA, so perhaps $g_S$ need not be small?
• In the black brane picture, the branes backreact so much that they create the geometry. This makes sense if $g_S$ is large. Also, the large $g_S$ argument seems to me to be at the heart of entropy calculations of black holes.

Or is the whole point that (some) branes are BPS objects? And so even if originally the SUGRA solutions were at small $g_S$ (since it is SUGRA we're talking about), they are protected objects and we may follow them at strong coupling too, and so effectively the black brane picture is valid at any $g_S$?

Many thanks.

asked May 24
edited May 24

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### Home > CALC > Chapter 11 > Lesson 11.1.3 > Problem11-31

11-31.

The position of a ball at time $t$ is specified by the point $(x(t)$, $y(t))$ where $x(t) = 20t$ and $y(t) = 40t-10t^2$. If distance is measured in meters and time is in seconds,

1. When will the ball land?

This occurs when $y(t) = 0$.

2. How far does the ball travel horizontally?

Use your answer from part (a) to evaluate $x(t)$.

3. How high does the ball go?

If you graph only $y(t)$, what is the shape of the graph? Where does the max occur on this type of graph?


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Mathematics

# Uncountable Sets

1.1k views

 1 Introduction 2 Further Examples 3 Properties of Number Sets 4 FAQs

21 September 2020

## Introduction

A set is uncountable if it contains so many elements that they cannot be put in one-to-one correspondence with the set of natural numbers. In other words, there is no way that one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

Uncountable is in contrast to countably infinite or countable.

The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.

For example, the set of real numbers between 0 and 1 is an uncountable set because no matter what, you'll always have at least one number that is not included in the set. This set does not have a one-to-one correspondence with the set of natural numbers. The proof of this involves creating an infinite list of numbers between 0 and 1 such as this.

 $$0.12348…$$ $$0.34897…$$ $$0.98789…$$ $$0.43238…$$ $$0.55349...$$

No matter what kind of list you create, there will always be a number that is not in the list. This is found by using Cantor's diagonal argument, where you create a new number by taking the diagonal components of the list and adding 1 to each. So, you take the first place after the decimal in the first number and add one to it. You get $$1 + 1 = 2.$$ Then you take the second place after the decimal in the second number and add 1 to it $$(4 + 1 = 5).$$ And so on to get your new number:

$$0.25840...$$

(When your number is a $$9,$$ you get $$0$$ when adding a $$1$$)

## Further Examples of Uncountable Sets

As we have already seen for countable sets, the concept of countability and cardinality will be explained through examples:

1. Rational Numbers (https://cuemath.com/learn/Mathematics/rational-numbers/)

2. Irrational Numbers (https://www.cuemath.com/numbers/irrational-numbers/)

3. Real Numbers (https://cuemath.com/learn/Mathematics/real-numbers/)

4. Complex Numbers (https://cuemath.com/learn/Mathematics/complex-numbers/)

5. Imaginary Numbers

## Properties of the Number Sets

1. Closed under addition (multiplication, subtraction, division) means the sum (product, difference, quotient) of any two numbers in the set is also in the set.

2. Dense: Between any two numbers there is another number in the set.

3. Continuous with no gaps. Every sequence that keeps getting closer together will converge to a limit in the set.

4. Every polynomial with coefficients in the set has a root in the set.

## What is a countable set?

A countable set is a set of numbers that can have a one to one mapping with the set of natural numbers i.e. are either finite or countably infinite.

## What is an uncountable set?

An uncountable set is a set of numbers that don’t have a one to one mapping with the set of natural numbers i.e. they consists of infinite numbers.

## What are a few known examples of a countable set?

Examples of countable set include:

1. Natural Numbers
2. Even Numbers
3. Odd Numbers
4. Whole Numbers
5. Integers
6. Positive Integers
7. Negative Integers, etc.

## What are a few known examples of an uncountable set?

Examples of uncountable set include:

1. Rational Numbers
2. Irrational Numbers
3. Real Numbers
4. Complex Numbers
5. Imaginary Numbers, etc.

GIVE YOUR CHILD THE CUEMATH EDGE
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text:
Large Sky Area Multi-Object Fiber Spectroscopic Telescope

### DR2 UPDATE (20151215)

Several problems have been found and fixed since last DR2 update. All of the spectra and catalogs here have been modified according to the latest updates, which are shown as follows:

1) 11 fibers’ IDs were found being wrong. We investigated all of the sky spectra assigned to these 11 fibers and finally determined the date when those mistakes happened. Among them, IDs of 9 fibers were found having been wrong since the begining of survey in 2011, and 2 of them happened during the replacing of bad fiber units in the summer of 2013.

We have fixed this problem by reassembling the affected spectra to the corresponding parameters, both in the catalogues and fits files. Total number of affected spectra is 7758, including 6640 changed and added, and 1118 removed by the flags such as position_error, etc. Here we provide a list of affected fibers and lists of changed and removed spectra.

2) There were some mistakes in calculating MJD (both fits header and catalogs), so we modified these values of MJD in this new version. Also we note that all fits files (spec) are named as spec-MMMMM-YYYY_spXX-FFF.fits where ‘MMMMM’ represents the local modified Julian day (LMJD).

3) The velocity of some M star spectral templates was found incorrect, which were applied to calculate the radial velosity of M stars. Therefore, ‘rv’ and ‘rv_err’ of all M stars have been recalculated with Hammer M star spectral templates in this update.

4) Minor fixes to the value of keywords ‘objtype’ and ‘tcomment’ of certain spectra in the catalogues.

5) Supplement 566 spectra of stars, which were mistakenly omitted by eye-check due to the plot problem in original DR2. Here we give a list of these stars.

6) Supplement 295 spectra of galaxies, which was the results of redoing eye-check required by users. Here we give a list of these galaxies.

7) Remove 4666 spectra, which were flagged as ‘fiber position error’, but being ignored by pipeline during the early days survey. Here is a list of these spectra.


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# compact

Displays or alters the compression of files or directories on NTFS partitions. If used without parameters, compact displays the compression state of the current directory and any files it contains.

## Syntax

``````compact [/C | /U] [/S[:dir]] [/A] [/I] [/F] [/Q] [/EXE[:algorithm]] [/CompactOs[:option] [/windir:dir]] [filename [...]]
``````

### Parameters

Parameter Description
/c Compresses the specified directory or file. Directories are marked so any files added afterwards are compressed, unless the /EXE parameter is specified.
/u Uncompresses the specified directory or file. Directories are marked so any files added afterwards aren't compressed. If the /EXE parameter is specified, only files compressed as executables are uncompressed; if you don't specify the /EXE parameter, only NTFS compressed files are uncompressed.
/s`[:<dir>]` Performs the chosen operation on files in the specified directory and all subdirectories. By default, the current directory is used as the `<dir>` value.
/a Displays hidden or system files. By default, these files aren't included.
/i Continues performing the specified operation, ignoring errors. By default, this command stops when an error is encountered.
/f Forces compression or uncompression of the specified directory or file. Already-compressed files are skipped by default. The /f parameter is used in the case of a file that was partly compressed when the operation was interrupted by a system crash. To force the file to be compressed in its entirety, use the /c and /f parameters and specify the partially compressed file.
/q Reports only the most essential information.
/EXE Uses compression optimized for executable files that are read frequently, but not modified. Supported algorithms are:
• XPRESS4K (fastest and default value)
• XPRESS8K
• XPRESS16K
• LZX (most compact)
/CompactOs Sets or queries the system's compression state. Supported options are:
• query - Queries the system's Compact state.
• always - Compresses all operating system binaries and sets the system state to Compact, which remains unless administrator changes it.
• never - Uncompresses all operating system binaries and sets the system state to non-Compact, which remains unless administrator changes it.
/windir Used with the /CompactOs:query parameter, when querying the offline operating system. Specifies the directory where Windows is installed.
`<filename>` Specifies a pattern, file, or directory. You can use multiple file names, and the * and ? wildcard characters.
/? Displays help at the command prompt.

#### Remarks

• This command is the command-line version of the NTFS file system compression feature. The compression state of a directory indicates whether files are automatically compressed when they are added to the directory. Setting the compression state of a directory does not necessarily change the compression state of files that are already in the directory.

• You can't use this command to read, write, or mount volumes compressed using DriveSpace or DoubleSpace. You also can't use this command to compress file allocation table (FAT) or FAT32 partitions.

## Examples

To set the compression state of the current directory, its subdirectories, and existing files, type:

``````compact /c /s
``````

To set the compression state of files and subdirectories within the current directory, without altering the compression state of the current directory itself, type:

``````compact /c /s *.*
``````

To compress a volume, from the root directory of the volume, type:

``````compact /c /i /s:\
``````

Note

This example sets the compression state of all directories (including the root directory on the volume) and compresses every file on the volume. The /i parameter prevents error messages from interrupting the compression process.

To compress all files with the .bmp file name extension in the \tmp directory and all subdirectories of \tmp, without modifying the compressed attribute of the directories, type:

``````compact /c /s:\tmp *.bmp
``````

To force complete compression of the file zebra.bmp, which was partially compressed during a system crash, type:

``````compact /c /f zebra.bmp
``````

To remove the compressed attribute from the directory c:\tmp, without changing the compression state of any files in that directory, type:

``````compact /u c:\tmp
``````


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# Electrical – Causes for Saturation in Transimpedance Amplifier Circuit

operational-amplifierphotometrytransimpedancevoltage-clipping

I have a transimpedance amplifier circuit with a Silicon Photomultiplier as shown below:

simulate this circuit – Schematic created using CircuitLab

We're using the SensL MicroFJ-60035 Silicon Photomultiplier and the AD8014 amplifier. My team shines a light pulse on the SiPM, and the transimpedance amplifier converts the current signal into a voltage signal which we read through the coaxial connector. The rise time of the current signal is approximately 20 ns.

However, we are experiencing clipping at 1.5 V at the output. Sorry, I don't have an image to show here right now. This type of behavior seems strange, and I was wondering what could be the issue that's causing it? I'm not an expert on amplifier behavior, but could something at the input or the op-amp parameters be causing the clipping?

Based on my knowledge, clipping only occurs when you try to drive a signal above the power supply limits, so I don't really know what to look for in terms of the cause. I'm don't know if the rise time directly corresponds to the input signal frequency, but we are achieving around 6-10 ns rise time of our signal, so assuming that we're working with a frequency range of 100 to 200 MHz, looking at the graphs in the AD8014 datasheet, could we be operating a non-linear region which could clip the output signal? The AD8014 has a large bandwidth of 400 MHz, so I would think that it's large enough for our needs.

EDIT: Based on the comments, there's something I wanted to note here that may help. Previously, my team used the OPA656 from Texas Instruments for our op-amp. Once we found that it was too slow for our needs, I looked at other op-amps with a faster slew rate. Thus, I went with the AD8014. It has a faster slew rate, and it has the same power source requirements as the OPA656, so I thought that it would be a viable substitute. We never had issues with the OPA656, but with all these comments, I'm trying to look and see how else they are different so that I may understand why we're having problems with the AD8014.

EDIT: This is a revised edit of a previous edit that I made regarding a simulation. Some tests that I ran show some things more clearly, so I wanted to note it here. First off, below is my simulation schematic that I'm using:

In this schematic, I'm using SPICE models provided by the appropriate manufacturers for the SiPM and the op-amp. In this case, the OPA656 is shown here. This is important for the next part, but please keep in mind that I'm comparing the behavior of the OPA656 and the AD8014 in the same circuit. I also have a model for that as well. V4 is the voltage source that 'triggers' the SiPM. It pulses once after a 10 ns delay, and it stays on for 200 ps. The SiPM turns on, and the current signal gets turned into a voltage signal, and it goes into a 0.6 meter transmission line model. I used stats for the RG-58 coaxial cable going to a 50-ohm load.

Below is the simulation results showcasing the voltage at the op-amp output, the start of the transmission line, and the middle and end of the transmission line as well as the current through the end load.

It's a little hard to see, but I hope that the voltages are easy to understand. The current through the load resistor is at the top, and it goes up to a max current of around 17 mA. The voltage at the op-amp output reaches a peak of 1.7 V while the voltages at the line and load reach around 0.85 mV.

Below is the simulation result when using the AD8014 amplifier in the exact same circuit:

Right now, the team isn't so much concerned about the amplitude of the voltage but more of its shaping. With a feedback resistor of 50 ohms, you can see a sort of 'double-peak' here with the AD8014 that isn't present in the OPA656. The OPA656 goes into a small peak only once where with the AD8014, you see it rises to a peak, and it doesn't fall immediately but rather levels off kind of like there's a second peak.

I not only see this type of behavior at 50 ohms, but I also see some similar behavior at larger resistances as well. Below are images of the OPA656 and the AD8014 when the feedback resistor is 470 ohms, respectively:

Again, you can see here how at the load, the AD8014 reaches a peak, goes down, and levels off in a sense. You don't see it here with the OPA656. I'm not familiar with behavior like this, but what would cause this kind of pulse shaping to occur among op-amp circuits?

Regarding AD8014 output characteristics, from the data sheet. This with +/-5V power supply:

In your transimpedance circuit R3, a 235 ohm resistor appears to the output as a load resistor to ground. Its parallel 3pF capacitor also loads the output, but is nearly insignificant compared to the resistive loading.
R4, a 50 ohm series resistor is assumed to be closely associated with the output pin of AD8014. It would drive a 50 ohm coaxial cable to provide an output signal. That coaxial cable should see a 50 ohm load resistance at its far end, to prevent pulse reflections. R4 in series with this 50 ohm load would be seen as 100 ohms to the AD8014 output.
Total load at the AD8014 output would be 235 ohms in parallel with 100 ohms: about 70 ohms. Since data sheet (above) stops at 150 ohm load, it is somewhat dangerous to extrapolate down to 70 ohms. It appears that at 150 ohms, output stage requires a voltage overhead of 1.6 volts...(5V - 3.4V)....certainly when loaded with 70 ohms, overhead will be higher. It would seem reasonable that the output stage enters voltage limiting somewhere above +3V. Since R4 forms a voltage divider with the far-end 50 ohm load, half of this 3V limit would be all you could expect with a Vcc=+5V.

Output swing also has a current limit of 40 mA. This means that a total load resistance any lower than 75 ohms would limit output to +3V.

Your design is very close to voltage limiting, and to current limiting. If you want to increase these limits, you must raise Vcc and you must find a way to raise the total load resistance.

Edit Every op-amp's output has voltage overhead (even rail-to-rail has a little). Voltage swing is limited by this overhead, which subtracts from the DC supply voltage. Overhead voltage is weakly dependent on current, just as transistor saturation voltage is dependent on current flowing through the transistor.
Lightly loaded with 1k ohm, overhead is 1.4V. With a +5V supply, output voltage cannot reliably rise above 3.6V.
Loaded more heavily (150 ohm) overhead increases to 1.6V...output voltage cannot reliably rise above 3.4V. This is with a supply of +/-5V. Since this overhead subtracts from supply voltage, output swing can be increased by increasing supply voltage. Since your photodiode circuit swings only positive, it should be possible to improve voltage limiting by increasing +ve supply voltage (and perhaps decrease -ve supply voltage: you have an absolute maximum supply limit of 12V).

It appears that Analog Devices designs this device for a 50 ohm load, and specifies that output will reach +2V with such a load, drawing 40 mA. The spec suggests that output current above 50 mA should not be expected. Hard current limiting occurs at 70 mA.
Combining voltage swing limit with current limit sets the minimum resistance seen by the output pin: $$\ \frac {3.4}{0.05} \$$ = 68 ohms. This is very close to your current design. If you raise supply voltage to increase voltage swing, current limiting still applies....raising load resistance will be required to reduce current.
R4 need not be 50 ohms. It is most important to match cable impedance with an equal load resistance at its far (oscilloscope) end. However, lowering R4 will load the op-amp more heavily, and you run afoul of current limiting. Raising R4 to 470 ohm would result in far less voltage at the oscilloscope end, which must terminate with 50 ohms.
If you must connect a load via coax cable, a choice of RG-59 (75 ohm) or RG-62 (93 ohm) would be better, naturally with an appropriate termination resistor value. OPA656 has slightly stronger output stage - if this gave acceptable output, then raising output resistance this way should be good. With 75 ohm coax, you can reduce R4 to 25 ohms. If this is not an option, then this device is too feeble to supply more output signal.


text:
## Files in this item

FilesDescriptionFormat

application/vnd.openxmlformats-officedocument.presentationml.presentation

591604.pptx (4MB)
PresentationMicrosoft PowerPoint 2007

application/pdf

1612.pdf (17kB)
AbstractPDF

## Description

 Title: High harmonic generation XUV spectroscopy for studying ultrafast photophysics of coordination complexes Author(s): Ryland, Elizabeth S. Contributor(s): Vura-Weis, Josh; Verkamp, Max A.; Lin, Ming-Fu Subject(s): Instrument/Technique Demonstration Abstract: \newcommand{\midtilde}{\raisebox{-0.25\baselineskip}{\textasciitilde}} Extreme ultraviolet (XUV) spectroscopy is an inner shell technique that probes the M$_{2,3}$-edge excitation of atoms. Absorption of the XUV photon causes a 3p$\rightarrow$3d transition, the energy and shape of which is directly related to the element and ligand environment. This technique is thus element-, oxidation state-, spin state-, and ligand field specific. A process called high-harmonic generation (HHG) enables the production of ultrashort (\midtilde20fs) pulses of collimated XUV photons in a tabletop instrument. This allows transient XUV spectroscopy to be conducted as an in-lab experiment, where it was previously only possible at accelerator-based light sources. Additionally, ultrashort pulses provide the capability for unprecedented time resolution (\midtilde70fs IRF). This technique has the capacity to serve a pivotal role in the study of electron and energy transfer processes in materials and chemical biology. I will present the XUV transient absorption instrument we have built over the past two years, along with preliminary data and simulations of the M$_{2,3}$-edge absorption data of a battery of small inorganic molecules to demonstrate the high specificity of this ultrafast tabletop technique. Issue Date: 2016-06-21 Publisher: International Symposium on Molecular Spectroscopy Genre: Conference Paper / Presentation Type: Text Language: En URI: http://hdl.handle.net/2142/91116 Rights Information: Copyright 2016 by the authors Date Available in IDEALS: 2017-01-26


text:
## Algebra 2 (1st Edition)

Plugging in the given value of d and using order of operations to solve: $$= (6+1)^2 -6 \\ = 7^2 -6 \\ =49-6 \\=43$$


text:
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Diffraction grating single slit grating - historatose.ru

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#### Diffraction - Wikipedia

Diffraction from a three-dimensional periodic structure such as atoms in a crystal is called Bragg diffraction. It is similar to what occurs when waves are scattered from a diffraction grating. Bragg diffraction is a consequence of interference between waves reflecting from different crystal planes.

#### Diffraction Grating (Theory) : Optics Virtual Lab ...

Spectrometer, diffraction grating element and mercury vapor lamp. Theory: When a wave train strikes an obstacle, the light ray will bend at the corners and edges of it, which causes the spreading of light waves into the geometrical shadow of the obstacle. This phenomenon is termed as diffraction. Single slit diffraction: When waves pass through a gap, which is about as wide as the wavelength ...

#### Diffraction Gratings - HORIBA

Diffraction gratings are manufactured either classically with the use of a ruling engine by burnishing grooves with a diamond stylus or holographically with the use of interference fringes generated at the intersection of two laser beams (for more details see Diffraction Gratings Ruled & Holographic Handbook). Classically ruled gratings may be plano or concave and possess grooves each parallel ...

#### Diffraction grating (video) | Khan Academy

We call this a diffraction grating. So this is a diffraction grating and it's more useful than a double slit in many ways because it gives you clearly delineated dots and it let's you see them more clearly. How many holes are there in a diffraction grating? Well typically, these are rated in lines per centimeter. In other words, if you took a diffraction grating and you asked, how many holes ...

#### Diffraction Gratings: Theory and Applications

Diffraction Gratings: Theory and Applications 9 Applications zReflective Gratings are wavelength-selective filters. Other examples of filters are Fiber Bragg grating, Fabry-Perot, Mach-Zehnder, etc zIn optical communications, they are used for 1. Wavelength Selection: Splitting and/or combining optical signals 2. Pulse Compression: Normally as reflectors in external cavity DBR lasers. Patrick ...

#### Diffraction Grating : Hitachi High-Tech GLOBAL

Diffraction gratings for compact, high efficiency monochromator: These gratings are designed for normal incidence mounting. They reduce aberrations and at the same time increase the efficiency of light intensity. A as a result, a high quality of image focusing of the concave grating is available. Diffraction gratings for flat field spectrograph: Variable space grooves enable flat-field image ...

#### Diffraction Grating - Definition, Formula and Solved Examples

Diffraction grating formula. Where, n is the order of grating, d is the distance between two fringes or spectra; λ is the wavelength of light; θ is the angle to maxima; Solved Examples. Question 1: A diffraction grating is of width 5 cm and produces a deviation of 30 0 in the second-order with the light of wavelength 580 nm. Find the slit ...

#### Diffraction Grating - Bergische Universität Wuppertal

Diffraction Grating. When there is a need to separate light of different wavelengths with high resolution, then a diffraction grating is most often the tool of choice. This "super prism" aspect of the diffraction grating leads to application for measuring atomic spectra in both laboratory instruments and telescopes. A large number of parallel, closely spaced slits constitutes a diffraction ...

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#### grating | Übersetzung Englisch-Deutsch

grating: Beugungsgitter {n} grating: Bildraster {n} grating: Rasterung {f} 2 Wörter: Substantive: amplitude grating: Amplitudengitter {n} optics phys. blazed grating: Blazegitter {n} mineral. crystal grating: Kristallgitter {n} optics phys. diffraction grating: Beugungsgitter {n} optics phys. diffraction grating: optisches Gitter {n ...

#### diffraction grating - LEO: Übersetzung im Englisch ...

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#### Diffraction grating | Light waves | Physics | Khan Academy ...

What happens when there's way more then two holes? Created by David SantoPietro. Watch the next lesson: https://www.khanacademy.org/science/physics/light-wav...

#### diffraction grating experiment - Demandpapers

June 3, 2017 0 0 . diffraction grating experiment diffraction grating experiment The Discussion of Results section includes an explanation of how the collected data provide logical and reasonable support for the statement found in the Conclusion. The Discussion of Results should be clear, specific, and reasonable. It is often a lengthy section of several […]

#### Where to buy diffraction gratings, reflective ...

Diffraction gratings are optical devices containing a periodic phase or amplitude structure. They may be used for grading spectrometers and monochromators or for the stretching and compression of ultrashort pulses. E: See also our encyclopedia article on diffraction gratings! Related articles: Bragg gratings, volume Bragg gratings, spectrometers, chromatic dispersion, pulse compression, pulse ...

#### OSA | Diffraction by perfect and imperfect amplitude grating

The paper presents theoretical formulas for calculation of diffraction by perfect infinite and finite amplitude gratings with Fresnel and Fraunhofer approximations. Further, general formulas for diffraction by an imperfect diffraction grating are derived where edges of the grating are described with general harmonic functions. Such a formalism provides enough power to accurately characterize ...

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Sep 26, 2020 - Explore Doug's board "Diffraction Grating" on Pinterest. See more ideas about Diffraction grating, Sculpture painting, Spiral galaxy.

#### diffraction grating - Deutsch-Übersetzung – Linguee Wörterbuch

The diffraction grating is a sine-phase grating of high resolution (2400 grooves per mm), manufactured by holography. nanoscale.de. nanoscale.de. Das Beugungsgitter ist ein holografisch hergestelltes hochauflösendes Sinusphasen-Beugungsgitter (2400 Teilungsperioden [...] pro mm). nanoscale.de . nanoscale.de. The crucial component here is the Fraunhofer IPMS-developed [...] micro scanning ...

#### Reflective Ruled Diffraction Gratings | Edmund Optics

A diffraction grating is an optical component that separates (diffracts) polychromatic (white) light into its component wavelengths. Each grating is fabricated from a highly accurate master grating that is copied many times. The duplication process is described below for replicated gratings. These high-quality instrument-grade ruled gratings will satisfy almost all of your diffraction ...

#### Diffraction grating (DGTD) – Lumerical Support

Characterize a diffraction grating in response to a broadband planewave at normal incidence. Lumerical provides a set of grating scripts for the DGTD solver, making it easy to calculate common results such as number of grating orders, diffraction angles and grating efficiencies at different wavelengths. download example . Overview. Understand the simulation workflow and key results. The ...

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A diffraction grating as claimed in any preceding claim, wherein said line of discrete apertures or wells is a straight line, and said reflecting facets are oriented such that their normals are aligned with the direction of linear extent of the grating. 7. A monolithic optical multiplexer or demultiplexer having as a dispersive element thereof a reflective type diffraction grating as claimed ...

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text:
# Tag Info

14

There is no difference. Let's look at how the output is constructed by adding \showoutput and checking the .log when compiling the following minimal example: \documentclass{article} \pagestyle{empty}% To avoid page numbers in the output \showoutput% Show how the output is constructed \begin{document} $xy=1$ $x y = 1$ \end{document} This is extracted from ...

12

There is no difference between $xy=1$ and $x y = 1$ or $x y = 1$. They are rendered the same exactly: The spacing depends on the classification into math atoms: $\mathord{x} \mathord{y} \mathrel{=} \mathord{1}$ Spacing: There is not spacing at the begin or end of the formula except for \mathsurround, which is usually 0 pt. No space between ...

8

At the beginning of each tikzpicture, TikZ selects nullfont and sets \spaceskip to zero. Then, in each node content, TikZ resets \spaceskip to .3333em. Here is a solution: \documentclass{article} \usepackage{tikz} \spaceskip 2ex \tikzset{every node/.style={execute at begin node={\spaceskip 2ex}}} \begin{document} \noindent a b c\\ \tikz{% \node ...

8

LaTeX is so not designed for this kind of document. But if you must, must. \documentclass{article} \usepackage{graphicx} \begin{document} \setlength\unitlength{1cm} \vspace*{\fill} \noindent \begin{picture}(0,0)\bfseries\large \put(4,3){\includegraphics{house}} \put(0.5,5){\parbox{5cm}{\raggedright If you are feeling generous I'd like a house, ...

5

I'd probably use array for this but in any case the same \extrarowheight parameter can be used: \documentclass[12pt]{article} \usepackage{tabu} \begin{document} a \[ \begin{tabu} {c | c c c c c c} \firsthline f & a_{11} & a_{12} & a_{21} & a_{22} & b_1 & b_2 \\ \hline 1 & a_{11}^1 & a_{12}^1 & ...

5

The following is taken directly from Knuth's TeXbook (Chapter 3: Controlling TeX, p 8): When a space comes after a control word (an all-letter control sequence), it is ignored by TeX; i.e., it is not considered to be a "real" space belonging to the manuscript that is being typeset. But when a space comes after a control symbol, it's truly a space. ...

5

There are not many uses for it besides after control sequences or to ensure non extended spaces after periods that are not punctuation (but in these cases, a tie ~ would be better). The tie is defined in terms of \ : in Plain TeX it is \def~{\penalty\@M \ } % tie while in LaTeX we see \def~{\nobreakspace{}} ...

5

How about the following: \documentclass{beamer} \begin{document} \frame{ \begin{itemize} \item one \item two \newline \item three \end{itemize} } \end{document}

5

You add vertical space in the same way you'd do outside itemize; it's not very clear why you'd do it in the first place. The following code works also with beamer. \documentclass{article} \begin{document} \begin{itemize} \item one \item two \bigskip \item three \end{itemize} \end{document}

5

I would suggest using enumitem to create list to suit your needs: \documentclass{article} \usepackage{enumitem} \newlist{noitemize}{itemize}{1} \setlist[noitemize]{label={}, labelsep=0pt, leftmargin=0pt} \begin{document} \noindent Some reference text. \begin{itemize} \item foo \item[] bar \end{itemize} \begin{itemize}[label={}] \item foo ...

4

One non-tabular approach using the tabto package: \documentclass[11pt,a4paper]{scrreprt} \usepackage{tabto} \parindent0pt \parskip.5em \begin{document} \NumTabs{3} \noindent {\bfseries Book title} \tab{\bfseries Author's name} \tab{2010} This book is about some guy who wrote a book because he felt like a book would be a good thing to write.....etc etc ...

4

Expanding on my comment: acro's options regarding formatting accept code whose last macro may need a mandatory argument (if this shouldn't work for a formatting option I'd consider this an error on my part and would appreciate a message!). The contents of said argument will be the corresponding entry. So I would recommend using \DeclareAcronym{aureus}{ ...

4

You need to alter the value of \labelsep for the list. The default is 0.5em. This changes that to 1.5em: \documentclass{article} \usepackage{regexpatch} \usepackage{algpseudocode} \makeatletter \xpatchcmd{\algorithmic}{\labelsep 0.5em}{\labelsep 1.5em}{\typeout{Success!}}{\typeout{Oh dear!}} \makeatother \begin{document} \begin{algorithmic}[1] \State ...

2

Use it this way: \documentclass[a4paper]{scrartcl} \usepackage[absolute]{pst-abspos} \begin{document} \section{one} \pstPutAbs(5,-5){Hi} \newpage \section{two} \pstPutAbs(6,-5){Hi} \end{document} With [absolute] it is internally already set at the beginning of the first page. And with \newpage it is set on the following page.

2

A simple solution is to wrap the two proof trees in a minipage each. \documentclass{article} \usepackage{bussproofs} \begin{document} \begin{minipage}{.45\textwidth} \begin{prooftree} \AxiomC{$\Gamma\vdash B$} \RightLabel{K} \UnaryInfC{$\Gamma,A\vdash B$} \end{prooftree} \end{minipage} \begin{minipage}{.45\textwidth} \begin{prooftree} ...

1

You can define a bprooftree (boxed proof tree) that encloses the proof tree in a box as wide as the tree. \documentclass{article} \usepackage{bussproofs} \usepackage{lipsum} % just for the example \newenvironment{bprooftree} {\leavevmode\hbox\bgroup} {\DisplayProof\egroup} \begin{document} \lipsum*[2] \[ \begin{bprooftree} \AxiomC{$\Gamma\vdash B$} ...

1

prooftree is designed to set a proof in a centred, display environment, with appropriate vertical spacing. You don't want that so don't use the environment. Instead, use \DisplayProof to display the proofs in-line. This can be used to set proofs within text or side-by-side. I've used a regular center environment to display the two proofs centred and with ...

1

I have used the following which puts x -> y under the limit. \lim\limit_{x \to y}

1

You can use \makebox to specify the width of the blocks. As the other answer shows, are other box commands that will give you other options like text wrapping (see texdoc lshort). If you want the date flush right you can just put an \hfill before it. This will all be simpler to write if you define a macro that spaces and formats your headers so that you can ...

1

As cgnieder correctly commented: Using long-format=\textit instead of long-format=\itshape does the trick.

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# Determination of configuration of geometrical isomerism

Geometrical isomerism: The molecular formulas of fumaric acid (m.p. 287 °C) and maleic acid (m.p. 130 °C) are identical, but the configuration of functional groups surrounding the double bond varies. They differ in terms of their physical and, to some degree, chemical characteristics. This type of isomerism is known as geometrical isomerism.

The methods of determination of configuration are classified as: (I) Chemical methods and (II) Physical methods

## Chemical Methods

These methods include:

(a) Absolute method: This method is based on the following observations. (i) Functional groups in a cyclic compound located cis to each other can be converted into cyclic lactones, anhydrides, or amides. e.g., maleic acid containing two –COOH groups cis to each other forms anhydride easily. Hence, it can be identified as cis-(maleic acid). Similarly, in fumaric acid, the two –COOH groups are on the opposite side, it can not form anhydride easily. Hence, it can be identified as a trans-form. (ii) Cis-isomers can be synthesized from small rings, but trans isomers can not be synthesized from small rings.

(b) Through chemical reaction not affecting the configuration of the double bond: The synthesis of trisubstituted alkene of known configuration is possible by syn addition of organo-copper reagent to alkyne followed by alkylation.

(c) If we synthesize a product from a starting material of a known configuration, then the configuration of the product remains the same as that of the starting material.

(d) The stereoselective reaction helps to predict the configuration of the resulting product. One such stereoselective reaction is the Wittig reaction.

## Physical Methods

The geometrical isomers differ from each other in their physical properties which include:

(a) Boiling point, melting point, density, refractive index and dipole moment
(b) Acid strength
(c) UV-visible spectra
(d) Vibrational (IR-Raman) spectra
(e) NMR (1H, 13C both)
(f) X-ray, microwave spectra, and electron diffraction methods.

### (A) Boiling point, melting point, density, refractive index, and dipole moment

It is difficult to anticipate the isomer configuration using factors like boiling temperature, melting point, density, and refractive index. Dipole moment varies between the cis and trans isomers, sometimes being higher for the latter. The trans isomer has more symmetry than the cis, similarly. Trans typically has a greater melting point as a result. e.g.,

### (b)Acid Strength

The acid strength is strongly dependent on the configuration of the compound e.g. pKa of cis and trans isomers of crotonic acid.

The initial dissociation of the cis form (maleic acid) is more acidic than the trans form (fumaric acid), but the second proton’s acidity is reversed. The maleic acid conjugate anion’s intramolecular H-bonding is to blame for this. Once the proton has first been dissociated, it is more stable than fumaric. As in fumaric, the second dissociation of cis-two negative anion species near to one another is not advantageous (the negative species further away). As a result, in the second dissociation, the transform is more acidic than cis.

### (c) UV-visible Spectra

Two large groups are on the same side of the cis isomer. Because of this intense internal packing, the molecule is less stable and has less resonance energy than the trans isomer. The cis isomer has an absorption maximum at a somewhat shorter wavelength than the trans isomer because it is distorted and forced to be non-coplanar.

(d) Infrared and Raman Spectra: The difference in the IR spectra of the two isomers may be pointed out in the following regions.

1650 cm-1 (C = C), and 970 – 690 cm-1 (= C – H out of plane vibration).

Similar for trans 1,2-dichloroethylene, the dipole moment is zero, due to its symmetrical nature.

Cis-isomer shows no IR absorption but shows Raman absorption at 1577 cm–1. While trans-isomer shows strong IR absorption at 1590 cm–1 but shows no Raman absorption.

### (e) NMR Spectra

NMR spectra can also reveal information on the location and arrangement of atoms (the molecule’s environment), in addition to revealing which functional groups are present. NMR spectra can distinguish between chemically dissimilar protons. In disubstituted ethylene, RHC = CHR’, where R and R’ have very different effects on the olefinic protons’ magnetic surroundings. As a result, these protons encounter a resonance state at different field intensities. The hydrogen atoms are considered to be deshielded because the olefinic protons are often observed in the low field of the NMR spectrum.

The trans isomer is strongly coupled and hence has a coupling constant of 17-18 c.p.s. (cycles per second). While the coupling constant of the cis-isomer ranges from 8-11 c.p.s. Similarly, the difference in chemical shifts of cis-and trans-isomers may be used to identify the configuration of the isomer.

### (f) X-ray and Electron Diffraction

The most effective method for precise structural characterisation of crystalline substances is single-crystal X-ray diffraction. It displays the spatial atomic arrangement and creates a picture of the crystal’s interior structure. The primary source of data on the geometrical structure of molecules, including bond lengths, bond angles, flexible molecule conformations, and intermolecular interactions, is single crystal X-ray diffraction.

Also read: Asymmetric synthesis


text:
# Differential geometry question

• Nov 18th 2007, 07:11 AM
musicmental85
Differential geometry question
Hi,
I have lots of theory to work off but no examples so I cant figure out
what Im doing for this question

Consider a unit-radius cylinder with centre running along the x-axis
a) write down a coordinate patch which covers this cylinder
b) compute the shape operater
c) find the principal curvatures and vectors
d) find the Gaussian and mean curvatures

I know one of the coordinate patches we have studied is the mange
patch. How do I know if that is a viable option here?

If it isn't I have a lemma that states for a coordinate patch PHI of a
function f:M->R we have PHI_u(f)=df/du and PHI_v(f)=df/du. Does this
mean I need to write a function for the cylinder and differentiate it
to get a patch?

The shape operater is S_p(v)=-(TRIANGLE)_v. n where n(p)=PHI_u(p) X
PHI_v(p)/||PHI_u(p) X PHI_v(p) but as I cannot work out what the PHI
is supposed to be I am completely stuck at the moment and can work no
further:confused:
• Nov 30th 2007, 05:12 PM
Rebesques
Let's see...

For a coordinate patch try $C: \Phi(u,v)=(v,cosu,sinu), u\in[0,2\pi], v>0$. Now we can calculate the normal at the point $p=\Phi(u,v)$, $\eta(u,v)=\frac{1}{(\sqrt{1+v^2})}\left(0,cosu,sin u\right)=(\eta_1(u,v),\eta_2(u,v),\eta_3(u,v))$.

For $w=(w_1,w_2)\in T_p(C)$, we have $S_p(w)=-\nabla_w(\eta)=-\left((d\eta_1)\Big{|}_{(u,v)}(w),(d\eta_2)\Big{|} _{(u,v)}(w),(d\eta_3)\Big{|}_{(u,v)}(w)\right)$
if my memory is not failing me again :o. For the first one say, we have

$(d\eta_1)\Big{|}_{(u,v)}(w)=\frac{\partial \eta_1}{\partial u}w_1+\frac{\partial \eta_1}{\partial v}w_2$, and so on.

The principal curvatures are just the eigenvalues of $S$.

For the Gaussian and mean curvature, either use the known formulae that include the principal curvatures, or play smart and notice that the cyhlinder and the plane are isometric, so...
• Dec 3rd 2007, 07:44 AM
musicmental85
Ok so I can work out the actual question when given the patch but how do I pick a coordinate patch?
• Dec 7th 2007, 07:50 AM
Rebesques
By force or by experience. :)

For the cylinder, it is quite easy, as two variables are always on a circle for fixed height. So you get polar coordinates to express these.

Why don't you try giving parametrizations for the sphere and the torus? Should be good practice.


text:
# Tag Info

20

It's a long-running joke that CFD stands for "colorful fluid dynamics". Nevertheless, it is used -- and useful -- in a wide range of applications. I believe your discontent stems from not sufficiently distinguishing between two interconnected but different steps: creating a mathematical model of a physical process and solving it numerically. Let me ...

11

Reducing the kernel width $\sigma_m$ will usually reduce the condition number. However, kernel matrices can become singular, or close to singular, for any basis function or point distribution, provided the basis functions overlap. The reason for this is actually quite simple: The kernel matrix $K$ is singular when its determinant $\det(K)$ is zero. ...

11

I am not an expert in machine learning, but I can outline the considerations that are relevant. The numerical calculations in machine learning are generally linear algebra -- either solving linear systems or linear least squares. For both types of problems, there are well-known backward-stable methods, so I will assume you are using a backward-stable ...

10

I have recently been working on exactly this topic. You may want to take a look at our paper: http://arxiv.org/abs/1209.2364. Why are you interested in the runtime prediction of linear algebra routines? Do you intend to use the model for a certain purpose?

8

I've implemented this recently, basically it counts how many times each specific colour borders another colour to make up a frequency table. To generate an image, a random colour and position are selected and the rest of the image is built up from there. Results aren't very coherent, but they match the colour palette of the original. In order to make sure ...

7

You shouldn't get the impression from the cited question that there is no theory, such that brute force testing of "all methods" (whatever that means; it's easy to produce infinite sequences of methods that are hard to exclude a priori) is the only viable approach. The problem is that if you don't know enough about your problem to formulate a coherent ...

7

I think you are mixing a couple different ideas that are causing confusion. Yes, there are a wide variety of ways to discretize a given problem. Choosing an appropriate way may look like "voodoo" when you are learning these things in class, but when researchers choose them, they do so drawing on the combined experience of the field, as published in ...

7

So the way I went about formulating the problem was to essentially write the following equations: The state that will be estimated, which is defined as a column vector, is the following: $$w = [vec(A)^{T},\bar{N}^{T}]^{T}$$ where $\bar{N}$ is the unknown average $N$ vector and $vec(A)$ is the vectorization operator on the unknown matrix A. Based on the ...

6

If I understand the link of hardmath correctly, a "support vector machine" in its most simple form is just a glorious name for a linear function $f(x)=b^Tx+b_0$ dividing the space into two half spaces according to the sign of the values. And separating two given sets (with disjoint convex hulls for existence of a separating hyperplane). The original ...

6

As a future hint, use the double question mark ?? to pull the source of the function: Looking at the scipy sources this error message comes up if the internal parameter alpha_k is zero or None. This value in turn is tied to the internal Wolfe Line search algorithm. In particular it is called when the search doesn't find a better value along the search ...

6

I wrote a Python package called PyPGE. PyPGE is a Symbolic Regression implementation based on Prioritized Grammar Enumeration (1), not Evolutionary or Genetic Programming. It produces a deterministic Symbolic Regression algorithm. (1) Worm, Tony, and Kenneth Chiu. "Prioritized grammar enumeration: symbolic regression by dynamic programming." Proceedings of ...

5

I suppose, you right and your network is not that big to 100%-utilize the GPU. The bottle-neck here seems to be not the GPU itself, but the transfer rate between RAM and VRAM and here the difference between 750 Ti and 780 Ti is not that significant. You can try to improve the training speed by hiding the latency of memory transfer - you have to assure you ...

5

So bottom line is I don't see any comprehensive work on the use of AI in M&S as a whole, let's say having models that can learn how to produce new improved models using the existing models. There's definitely some work out there on this. This is the field of scientific machine learning. Currently there's three major paths that I'd break it into: Neural ...

4

Here's the code and the output of a solution that uses the development version of algopy for automatic differentiation. As someone else said in the comments, you could also try other python packages. Theano and statsmodels come to mind.

4

First, just to clarify things, you're talking about solving optimization problems of the form $\min \| x \|_{1}$ subject to $\| Ax - b \|_{2} \leq \delta$ and related forms, right? There are many different applications in which problems are formulated as the minimization of the 1-norm of a vector subject to linear or least squares constraints. An ...

4

This topic has been discussed at some length on Cross Validated (aka stats.stackexchange) and Reddit: Why is Newton's method not widely used in machine learning? (see in particular Nick Alger's answer) Why use gradient descent with neural networks? L-BFGS and neural nets Why second order SGD convergence methods are unpopular for deep learning? How does the ...

4

The general idea that you have of learning an easy to compute model from results of your detailed simulation model and then optimizing the easy to compute model is long-established. The easy to compute model is typically called a surrogate model or a response surface model. Once the surrogate is available, you can use conventional optimization techniques ...

3

I think I can address the second part of your Question. The phrase "sum-normalized to zero" is a fancy way of saying "subtract the mean (average)", i.e. subtract the constant needed to give a zero sum over the resulting function (filter) values. The phrase "square-normalized to 1" applies to the result of the first phrase and means dividing by a (...

3

A couple of suggestions: Choose $\sigma \sim$ the average distance | random $x$ - nearest $x_i$. (A cheap approximation for $N$ points uniformly distributed in the unit cube in $\mathbb{R}^d, d\ 2 .. 5$, is 0.5 / $N^{1/d}$.) We want $\phi( |x - x_i| )$ to be large for $x_i$ near $x$, small for background noise; plot that for a few random $x$. Shift $K$ ...

3

Sounds too simple so maybe I've misunderstood you, but they have binary variables, hence obviously a nonconvex problem (a binary variable is either 0 or 1, which is a nonconvex set)

3

The neural net in the second article doesn't use linear activation functions. It uses thresholded on-off functions (e.g. f(x) = 1 if x > threshold, else 0), hence can model XOR. (A linear activation function would just be f(x) = x).

3

If I interpret the problem right, the task is to assign to each $j$ in the column index set $J$ of the distance matrix an index $x_j$ from the row index set $I$ such that the $x_j$ are all different and the $\sum_j d(x_j,j)$ is minimal. This can be posed as a combinatorial constraint satisfaction problem to a number of constraint solvers. See, e.g., http://...

3

After a cursory google search on the subject, it appears that "symbolic regression" is a problem that lends itself greatly to stochastic optimization algorithms like genetic programming (GP). It is conceivable that you should look for an open source genetic programming library with modules specifically for symbolic regression, such as DEAP (Distributed ...

3

I once started writing anopen source version of Eureqa in Java. The project has limited capabilities but it implements the fitness function described in [1] and couple optimizations mentioned by the authors in other publications (e.g., searching for solutions in Pareto front). Link: https://github.com/pkoperek/hubert [1] Schmidt, Michael, and Hod Lipson. "...

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The problem is not convex, as one can verify using the simplest possible choice: $N=2, d=1$ and with data $x_1=x_2=1, f_1=1, f_2=0$. In that case, your objective function is $$F(C) = 2\left(1-\exp(-C/2)\right)^2.$$ Plotting this function shows that it is not convex.

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I learn computation science through Practical Numerical Methods with Python. https://github.com/numerical-mooc/numerical-mooc/wiki It covers finite differencing and many other numerical algorithms. For me the most interesting part is that it shows how various factors impact the stability, accuracy, and performance through small working examples.

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This metric is pretty much as misleading (or useful, depending on your perspective) for GPUs as it is for CPUs. Currently, a lot of applications/algorithm's implementations are limited more by memory throughput rather than FLOPs. Memory throughput (in GB/s) is also always listed for GPU specification and those two numbers together give a much better view on ...

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I have limited experience in machine learning; however, simplifying, you can think of it as a "trained black box". I would say, you have to know a lot about your problem, the behaviour of your functions in order to successfully and reliably apply any form of machine learning. You would have to decide on how to get the training data, size of the training set, ...

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# Math Help - I need formal proof to a sentence

1. ## I need formal proof to a sentence

can someone give the proof to this:
a*a + ab + b*b > 0
for a =/= 0 , b=/= 0.

thanks

2. A square number is always positive, so you have a^2 + b^2 > 0

No matter what a and b are, one number will be larger, so either a > b or b > a. So if for example if a < b, then b^2 will always be larger than ab, as firstly a square number is always positive, and secondly b x b is larger than a x b when a < b. So this means that the a^2 or b^2 will always be higher than ab can possibly be, so the number will always be above 0, then adding another square number cannot decrease the number, so adding the other value^2 will not bring it below 0.

My proof is rather messy, but I think you get the idea.

3. Another way :

$a^2 + ab + b^2 = \left(a+\frac{b}{2}\right)^2 + \frac{3}{4}b^2$
It is therefore the sum of 2 squares

If you prefer
$a^2 + ab + b^2 = \frac{(a+b)^2}{2}+ \frac{a^2}{2} + \frac{b^2}{2}$