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The basic idea of Gödel numbers is to establish a mapping between logical statements and natural numbers. This allows to apply reasoning on natural numbers, and those results can be translated back into reasoning on logical statements. Related mappings are: the mapping between Turing machines, one embodiment of computable functions, and natural numbers. ...

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Yes. The Information Age is based on the Gödel number. A program in the programming language Python, for instance, is stored internally in the computer as a base-128 integer, via the ASCII which encodes characters as integers. If you wish to execute the program, you provide it to the computer as a Gödel number, expressed in base 128 (by typing in certain ...

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The reason there is no Nobel Price for Mathematics is because Alfred Nobel’s fiancée wife had an affair with a mathematician. You’d be hard pressed to take any upper level university level mathematics course and not hear some variation of this story at least once per term from your mathematics professor (I personally have heard it from three different ...

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You might wanna check out Mathologer.

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If you'd like a super-summerized introduction you may want to check these course notes on singular homology : http://math.arizona.edu/~rwilliams/SingularHomology.pdf Again there is no proof or anything and it only covers the basic but I found it helpful just to, you know, see the big picture. Hope that helps.

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Have you ever felt self doubt in your career as a mathematician? How did you overcome those worries? Also, what are some good techniques or resources to advance one's skills as a undergraduate level mathematician? You bet I did. I graduated with a B.S. Mathematics, Statistics emphasis two years ago, and I am now pursuing a M.S. Statistics at a top-20 ...

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Im not studying and Im not going to study mathematics officially (Im a bit "old"), Im an amateur. When started to follow my hobby more seriously sometimes I feel that Im not understanding anything just memorizing things. But the brain is a giant mystery: one or two years after I started the "hobby", with long periods of time not seeing something about ...

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Even some of the most extraordinarily talented mathematicians, scientists, and scholars experience doubt--personal doubt that they've chosen the right career, worked on a solvable or important problem, that they can ever solve the problem (before a competitor) and so forth. This happens with every challenging discipline. Judge your progress both on a ...

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disclaimer: I'm also a sophomore in college studying mathematics. I've taken algebra, analysis I, measure theory, algebraic topology. Here are some things that helped me a great deal: (1) Spend a lot of time here, you get a sense for how proofs go, in some sense. It's very worthwhile to see how different people approach a singular problem. Answering ...

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Pythagoras' theorem! Proof using a rotated square within a square. This gets straight to the essence of mathematics!

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To prove: $0=1$. Certain identities get funky when we pass over to infinite-order matrices. We see such matrices, for example, in representations of operators in quantum mechanics. Everyone knows that $Tr(AB-BA)=Tr(AB)-Tr(BA)=0$. So let $A_{i,j}=\delta_{i,j-1}, B_{i,j}=A_{j,i}$ Here $\delta$ is the Kronecker delta function, and $i$ and $j$ run ...

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Let me try to say something as promised. First, an obvious point: slogans like this can never give the full picture. If you take this slogan completely literally, it would imply that if you knew the cone of curves, then you would know everything about the birational geometry of $X$. That would include, say, the whole birational automorphism group of $X$. ...

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I just finished my third year at college. I'm a math major with an additional major in statistics, and I feel like I kind of understand what you are talking about. I spent the first two years taking math core courses, and I had the same question as you do right now. I don't know what I can do with those math skills. Then I started to take some statistical ...

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Every axiomatic system begins with undefined concepts, definitions, and axioms. For example, in geometry, some of the undefined concepts are "point", "line", and "incidence". Likewise, in set theory, some undefined concepts might be "element", "set", and "is a member of". The peculiar thing about undefined concepts is that when you do a dictionary ...

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We can construct the initial algebra $\mathbb{N}$ of the category you describe as the set of all terms built from a nullary constructor (operation symbol) $\mathsf{Z}$ and a unary constructor $\mathsf{S}$. The initiality means that for any diagram of sets and functions $1\xrightarrow{z} X\xrightarrow{s} X$, there is a unique function $f\colon\mathbb{N}\to ... 2 As others have said, an augmented chain complex is an$(\mathbb{N} \cup \{-1\})$-graded chain complex$C_*$with$C_{-1} = \mathbb{Z}$(or more generally your base ring$\Bbbk$) and such that$H_{-1}(C) = 0$, equivalently the map$C_0 \to C_{-1} = \mathbb{Z}$is surjective. The splitting result follows from the projective property of$\mathbb{Z}$among ... 2 One considers$\mathbf Z$as a complex, the complex all of whose chain groups are$0$except the one in degree$0$where we have the group$\mathbf Z$. We write this complex as$\mathbf Z[0]$. An augmentation of a complex$C$is then an epimorphism of complexes$C\rightarrow \mathbf Z[0]$. A morphism of two augmented chain complexes$\delta\colon ...

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From some googling, apparently an "augmented chain complex" is defined to be a chain complex together with a surjective homomorphism $C_0\to\mathbb{Z}$ such that the composition $C_1\to C_0\to\mathbb{Z}$ is $0$. This meaning seems compatible with your homework problem. I must confess that I had never seen this definition before now; if someone asked me ...

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The distinction you introduced between vectors as arrays (columns or rows), on the one hand, and vectors as change, on the other is well taken. I would like to comment on vectors as change, used in fields ranging from calculus, differential geometry, and dynamical systems to mathematical physics. In this approach, a vector is thought of as an infinitesimal ...

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The most comprehensive treatment of residue-based techniques that I know is the two volume set: The Cauchy method of residues: theory and applications by Mitrinović and Kečkić, Dordrecht, 1984 (ISBN: 9027716234). The Cauchy method of residues: theory and applications, Vol. 2 by the same authors, and publisher. This one published in 1993 (ISBN: ...

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A vector space is a mathematical structure, and while in itself is quite informative, once you know a set is a vector space it doesn't mean your work is done, you can stop burning the midnight oil and hit the hay. For example: there's an obvious vector space isomorphism between the space of $n-1$ degree polynomials (call $\mathbb{R}[x]$) and ...

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The formal definition of a vector is pretty open ended (a member of a vector space). At a very high level vector is a collection of mathematical objects, that obeys rules of addition and scalar multiplication. A container of numbers isn't too bad. But the objects could be something like differential operators. And, they could be other vectors. But then, ...

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For the sake of an answer: This usually means that the solutions are less regular than the functions in the equations or the initial data.

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In algebra and number theory, cyclotomic fields $\mathbb{Q}(\zeta_n)$ with a primitive $n$-th root of unity $\zeta_n$ are very interesting. Their ring of integers is given by $\mathbb{Z}[\zeta_n]$, and is used for some cases of Fermat's last theorem. For example, for $n\le 19$ prime this ring has class number $1$, and hence is a factorial ring. This can be ...

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Let's assume we are in the context of a filtered probability space $(\Omega,(\mathcal F_t),\Bbb P)$ satisfying the usual conditions. Here are a few features of optionality that make it important. If $T$ is a stopping time and $A\in \mathcal F_T$ then there is a bounded optional process $Z$ such that $Z_T=1_A$ on $\{T<\infty\}$. If $Z$ is an optional ...

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Rudin's Principles of Mathematical Analysis and Axler's Linear Algebra Done Right

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a) Cohomology should be regarded as a graded-commutative algebra, therefore is a finer invariant. This allows for some quite interesting definitions, e.g. the Hopf invariant. b) If $M$ is a manifold of suitable dimension, then one has $b_{n-k}=b_k$ for the Betti numbers (proved via Poincare duality, which involves cohomology). It is sometimes easier to ...

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I think you've got the terminology mixed up a little. You have properties of a process (such as being adapted to a filtrartion, being progressively measurable, or being predictable), and then you have stopping and optional times (where the difference is, roughly speaking, merely that optional processes cannot answer questions regarding the "current time"). ...

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When I work with numbers in base 16 or base 2, the string "10" is not called "ten", but "one zero". We generally don't have names for specific integers that reflect some other base, so although a full 16 bit value is ffff and then 1 0000 is significant to me, I don't have a spoken name for it that corresponds to thousand. It's only known as "sixty ...

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The name refers to the way we choose to group our items. The way we've evolved, we found that nine counting symbols and a symbol for nothing suffice for our preferred base. We're able to recycle the glyphs $1$ and $0$ to denote our grouping, which is given by the combined symbol $10$. To illustrate why it's all about the grouping, and not about the final ...

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You're mixing the actual number with its representations. Yes, in base $n$, the string $10$ represents exactly $n$. But now forego of the decimal representation of $10$, and think about it as "how many digits a healthy human being has on both their hands". This is your $n$, now. Let's for the sake of simplicity call this number "ten". Now we are counting ...

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This is more like a comment but it's too long, so I'm putting it as an answer instead, please accept my apology. In base $10$, the "symbol" $78152_{10}$ represents the number $78152_{10}=7\cdot 10^4 + 8\cdot 10^3 + 1\cdot 10^2 + 5\cdot 10^1 + 2\cdot 10^0$. In base $n$, the "symbol" $78152_n$ represents the number $78152_n=7\cdot n^4 + 8\cdot n^3 + ... 4 The only reason the question seems silly is that you include the answer! A locally convex TVS is one that has a basis at the origin consisting of balanced absorbing convex sets. The reason for the emphasis on "convex" is that that's what distinguishes locally convex TVSs from other TVSs: every TVS has a local base consisting of balanced absorbing sets. ... 1 There is two different yet equivalent definition of Locally convex spaces : one in which the topology endowed by a family of semi-norms, and one in term of absorbent balanced and convex basis. The equivalence between the two definition is rather long to prove but you can find it in Rudin's Functional Analysis. This might be of interest to you : ... 0 If you are looking for a brief review of differential and integral calculus and are not too concerned with rigor a better site for you might be the Physics SE. Physicists tend to adopt a no-nonsense approach to the calculus that gets you there faster. Many mathematics courses often get bogged down in the gory details of the epsilon-delta approach so as to do ... 5 Note: We look at a few historical notes about the beginning of dimension theory in topology. We can find this way some arguments which led mathematicians to introduce new types of dimensions by means of functions. Then we take a look at fractal geometry and provide some information about desirable properties of dimensions. The following is a verbatim ... 0 Random by itself doesn't mean much. There are 'patterns' or distributions of random (uniform, normal, Poisson, etc). Each distribution is a way of assigning a 'likelihood' or probability to certain events occuring in an experiment. 0 A morphism is a concept introduced in the language of categories to designate one element of the set Hom (X, Y) where X and Y are two objects of said category. So if we talk about the category of sets, a morphism is just a mapping. if we talk about category of groups, a morphism is a group homomorphism ie mapping that complies with the laws of the groups in ... 0 Let$G$and$H$simply be sets filled with numbers. We can easily define mappings between$G$and$H$. Let's say that$f : G \to H$is given by$f(g) = g * g = g^2$. We could also say that$f(g) = g^3$or$f(g) = \frac{57}{g}$. These are all "mappings". A "mapping" is simply a rule that one uses to take an element of$G$and "mess around" with it to get ... 2 The best calculus textbook based on infinitesimals is Keisler's textbook Elementary Calculus. This opinion is based on our experience teaching calculus with infinitesimals based on the book over the past three years. We have taught over 250 students by now using this method, yielding better results than parallel groups that did not use infinitesimals, based ... 2 Hope this relevant to what was asked... To focus on$u = f(x), du = f'(x) dx,$take this as an easy way to write$u = f(x), \Delta u = f'(x) \Delta x,$where$\Delta x$is an interval in$x$that we can make arbitrarily small. This formula makes sense for$f$smooth because in a small enough region around$x,f$is locally linear. Recall that you can ... 2 The usual treatment of calculus today avoids using infinitesimals entirely.$\mathrm{d}x$is merely a notation that does not literally refer to an infinitesimal. That said, there is an alternative approach that uses infinitesimals known as nonstandard analysis. The Wikipedia articles on nonstandard analysis and nonstandard calculus are probably good places ... 5 In discrete time at least, the definitions I'm familiar with are fairly straightforward. Given an increasing filtration$\{\mathcal{F}_n\}_{n=0}^{\infty}$, a process$\{X_n\}_{n=0}^{\infty}$is adapted if each$X_n$is$\mathcal{F}_n$-measurable. For predictable processes, the random variables are measurable with respect to slightly smaller ... 0 I'm going to Georgia Tech this fall and here are a few things I included in my application that I believed helped me get accepted. -Take the SAT Math II subject test and shoot for a 750-800. It's not required my most schools but a very high grade in it shows promise in the subject. -If your school has a math club, join. If not, start one. That's what I did ... 2 If I am interpreting your question correctly, what you refer to as "a notion of time" is really just an order of summation. The key property of time that you are using is that it has a definite direction ("the arrow of time"), just like the order in which the partial sums are developed for a conditionally convergent series. ... 1 I have one problem that I have been working on for over ten years now. I work on it when a new idea occurs to me or when I get the desire to review my notes. I really can't leave it, but I am seeing other problems on the side. -6 I have a beautifully simple argument which will help you to show many doubters the true situation. If you project the real number line on to a circle it is clear that the real numbers are dense everywhere, while the natural numbers are dense only at infinity. This creates a strong, visualisable argument that there are more real numbers than integers. The ... 1 I found the assignments from the dead link previously posted. Here is the list in case the link dies again. HOMEWORK Assignment 1: Due in class, Thursday August 25, 2011 required: Chapter 1: 1, 2, 5, 6, 10, 11, 13, 14, 15, 20, 25 extra: Chapter 1: 16, 17, 22 Assignment 2: Due 4 pm, Friday September 2, 2011 required: Chapter 2: 1,2, 3 parts a ... 4 If you think of$p \vee q$as equivalent to$\neg p \implies q$and$\neg p \vee r$as equivalent to$p \implies r\$, this is used all the time, whenever you have a result that has two cases.

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My vote goes to Carl Friedrich Gauß: "Mathematical historian Eric Temple Bell said that had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years" (https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#Personality) See also Unpublished Discoveries by Gauss that Were Later Rediscovered and Attributed to ...

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