# All Questions

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### show that the sequence {a_n} is a cauchy sequence

Show that any sequence {a_n} that has the property |a_n+1 -a_n|< b^n for b<1 is a Cauchy sequence. I'm having problems giving a formal proof of why this holds.
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### How do you Compute $7^{1000} \mod 24$?

I'm being asked to compute $7^{1000} \mod 24$. I have Fermat's Little Theorem and Euler's Theorem. How do I use these to compute $7^{1000} \mod 24$? I'm stuck because $24$ is not prime. In this case, ...
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### Graph partitioning with constraints

Consider having $n$ data points ${{x}_{1}},..,{{x}_{N}}\in {{R}^{D}}$. Given an affinity matrix of the data, $W=[{{w}_{ij}}]$ where ${{w}_{ij}}$ is the affinity measure for data points ${{x}_{i}}$ and ...
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### Coherence theorem for symmetric monoidal categories

What's the formal statement for the coherence theorem for symmetric monoidal categories? I've seen there's some notion of permutation around, but I can't get my head around the thing that "all ...
I have four variables, $c_1, c_2, p_1, p_2$. I want to create two variables, $c_1, c_2$ around $p_1, p_2$. The ratio between the distances of $p_1, p_2$ and $c_1, c_2$, as $\beta = ... 0answers 12 views ### How can I solve this Diffrential equation? $$\frac{dy}{dx} = \frac{y+2}{x+1} + \tan(\frac{y-2x}{x+1})$$ How can I turn it into a seperable differential equation? (I just want a hint). 1answer 15 views ###$\frac{dy}{dx}=\cot(x)[\frac{\sin(x)}{y}-\frac{y}{2}]$Hello fellow Mathematicians , I am stuck with this problem here. $$\frac{dy}{dx}=\cot(x)[\frac{\sin(x)}{y}-\frac{y}{2}]$$ I did open bracket and it became ... 0answers 6 views ### relationship between the CNOT gate and the I and X single-qubit gates I need to prove this relationship:$G_{CNOT} = |0\rangle\langle 0| \otimes I + |1\rangle\langle 1|\otimes X$. So I think I need to show that both sides are a linear map on$H \otimes H \otimes H$So ... 1answer 24 views ###$(G\times G')/(H\times H')\simeq (G/H)\times(G'/H')$. Let$G_1$and$G_2$be two groups. Let$H_1$and$H_2$be normal subgroups of$G_1$and$G_2$respectively. Then prove that$(G1\times G2)/(H1\times H2)$is isomorphic to$(G1/H1)\times(G2/H2)$. ... 2answers 20 views ###$\frac{dy}{dx}+\sin(\frac{x+y}{2})=\sin(\frac{x-y}{2})$Hello Mathematicians I am stuck at a differential equation problem $$\frac{dy}{dx}+\sin(\frac{x+y}{2})=\sin(\frac{x-y}{2})$$ I don't know here where to begin with.In other kind of problem I have ... 0answers 13 views ### Using Lagrange form of the remainder with cosh I am trying to find "$\cosh 4$using the sixth partial sum ($n=5$) of the Maclaurin series" for the function. I am also trying to use "the Lagrange form of the remainder to estimate the number of ... 2answers 9 views ### how to impose binarity constraint in a vector This is part of a homework problem. In an optimization problem, I need to have a K dimensional vector S, such that each entry of the vector is either 0 or 1, and$l_1$norm of S is <= K. I can't ... 0answers 7 views ### Is there a program (webbased or not) that allows me to draw freeform on a graph? I need to draw some graphs for a distance-based course I'm taking where I need to draw graphs that are based on my own estimates, not on a function. All of the websites I'm finding online are based ... 2answers 28 views ### I made some mistake i cant see? so i got this limit: $$\lim_{x\to1}\frac{\sqrt[3]{x}-1}{\sqrt[4]{x}-1} \implies \lim_{x\to1}\frac{\frac{x-1}{\sqrt[3]{x²}+\sqrt[3]{x}+1}}{\sqrt[4]{x}-1} \implies ... 0answers 4 views ### Is this problem suited for Bayesian inference? Suppose that the quality of a widget is distributed according to a score, given by a normal distribution with mean 1 and variance σ^2. A fraction, π of all widgets are defective. The cost of having an ... 1answer 15 views ### What percentage is the rest of the black cars in the parking lot Ok so, Last Monday, a parking lot had 80 cars. Of those 80 cars, 25% were silver. 16 were red, and the rest were black. What percent of the cars were black 0answers 10 views ### I have trouble understanding the proof of the Wold decomposition theorem I'm trying to understand the proof of the Wold decomposition theorem in [1, p.187]. I find a few things about it very irritating. The theorem states: Theorem 5.7.1 (The Wold Decomposition). Let ... 1answer 11 views ### Probabilty & Statistics Problem The number of cracks which are present in a part of an international road has an average value of 2 cracks per kilometer. 1)What is the probability that there are no cracks in a section of road ... 3answers 15 views ### Analytical aptitude - Division of exponentials. What is the remainder when 6^17 + 117^6 is divided by 7? How to approach these type of questions? 0answers 5 views ### Primality of F_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1) , when q is prime, j\ge0? Let P_{q,j}=(2^{q^{j+1}}-1)/(2^{q^j}-1) , q prime and j\ge0. P_{2,j} is a Fermat number, P_{q,0} is a Mersenne number. Apart from Fermat primes and Mersenne primes, and apart from ... 2answers 30 views ### Prove or disprove : if f define on A \subset \mathbb R^n and its graph is connected, then f is continuous. Prove or disprove : if f define on A \subset \mathbb R^n and its graph is connected, then f is continuous. I don't think this is true, I know that if set A is path connected then A is ... 1answer 11 views ### What are good introductory books on the various math software packages? I feel like I've finally gotten a good grip on mathcad, and would now like to start going through the other popular software packages (Mathematica, Maple, and Matlab). Could anybody recommend general ... 0answers 10 views ### A simple question about positive element in C*-algebra I am reading a book about C*-algebra. There is a quotation below. An operator~system E is a closed self-adjoint subspace of a unital C*-algebra A such that 1_{A}\in E. The n \times n ... 0answers 8 views ### Notation for pointwise versus “setwise” stabilizers Suppose one is working with both pointwise and setwise stabilizers of sets under a group action. Are there common conventions for notationally distinguishing these two notions? How common are they? ... 0answers 9 views ### associated primes and finite base change Let R be an integrally closed commutative Noetherian integral domain. Let R \subseteq S be a ring extension such that S is also an integral domain and is finite as an R-module. Let I be an ... 2answers 12 views ### How do I calculate the new x y coordinate for a rectangle when centering it within a rectangle? I need to center a rectangle inside another rectangle. I know the width and height of the parent rectangle, and I know the width and height of the child rectangle that needs to be centered. I need ... 0answers 12 views ### Size issue in localization of a category. I have read on nlab and other sources that I can't backtrack that the localization process of a category can lead to size issue. More especially, starting from a locally small category \mathsf C and ... 1answer 13 views ### Integral of a differential 1-form along a curve (clarification on the definition) Let's denote with (e_1,\dots,e_d) the usual basis of \Bbb R^d, and with ({e_1}^*,\dots,{e_d}^*) the dual basis of its dual space \Bbb {(R^d)}^*. Let U be an open subset of \Bbb R^d and ... 1answer 23 views ### Lebesgue outer measure satisfies \lambda^{*}([a,b]) \leq b-a Aaagain, I fail to understand the trivial: Using compactness argument it is straightforward to show:$$\lambda^{*}([a,b]) \geq b-a$$And everything is OK. But, regarding \lambda^{*}([a,b]) \leq ... 2answers 58 views ### Is dxdy really a multiplication of dx and dy? In this link Is dy/dx not a ratio? it was told that \frac{dy}{dx} cannot be seen as a quotient, even though it looks like a fraction. My question is: does dxdy in the double integral represent a ... 0answers 19 views ### Which answer is true? Suppose \{E_n\} is a sequence of measurable sets in measure space (X,M,\mu) such that for every n, \mu(E_n)>\frac{1}{100}. For the set F = \{x \in X: x belongs to an infinite number ... 1answer 119 views ### On the evaluation of a beautiful series! Find value of \sum_{n=1}^\infty \left\{e-(1+\frac1n)^n\right\} exactly!? I found this question from here: 'Titu Andrescu-Problems in Real Analysis-p.114' 0answers 17 views ### Finding the elements of this subgroup I'm trying to find all of the elements of the subgroup of C* generated by (1+i)/Sqrt[2]. I know how to find elements in a subgroup in Z and U, but I don't know where to start here. 0answers 15 views ### An exercise in operator theory Let H be a Hilbert space and P be a projection to a finite dimensional subspace K of H, for a T\in B(H), if ||PTP||=1, then, for arbitrary \epsilon>0, there exists a vector \alpha ... 0answers 43 views ### Exact arctan value. Can't be solved ?? I am kind of stuck here. I need to know the exact value of$$ \arctan\left(\sqrt{2} \over 4\right). $$I am looking into double angle formula's, half angle formulas but honestly, I can't find it? Is ... 0answers 19 views ### Dummy variable in the probability generating function I'm struggling to understand what the purpose of the dummy variable t in the probability generating function is? I know it takes a value between 0 and 1, and have heard it described as a 'relative ... 1answer 14 views ### product of cumulative distribution function For all integer N>1, I am trying to show that for a gaussian (or even better any type) cumulative distribution function F(\theta;\mu,\sigma) (\mu and \sigma are the mean and standard ... 0answers 16 views ### Difference between sum of alternate terms of a decreasing series. I'm given a series say 5,7,8,4,2,10 First arrange terms is decreasing order i.e 10,8,7,5,4,2. Then calculate sum of numbers at alternate positions ie s_1=10+7+4 s_2=8+5+2 Required answer ... 0answers 40 views ### IEEE 754 as a mathematical space Integer operations in computers (i.e. 32-bit integers) probably can be represented best by modular arithmetic (because of integer overflows/underflows). What about IEEE 754 floating point arithmetic? ... 0answers 11 views ### completion of the canonical module For a local Noetherian Cohen-Macaulay ring (R,m,k) the canonical module is defined to be any maximal Cohen-Macaulay module of finite injective dimension and of type 1. The canonical module is ... 0answers 24 views ### Existence of unique periodic solution of ODE How to prove or disprove that the ODE$$ y'(x) = y( x )^9 + ( 1+\sin( x)) y(x) +\cos(x) =0$$has the unique periodic solution? PS. Its fieldplot done with Maple 3answers 54 views ### Proof:$a^2 - b^2 = (a-b)(a+b)$holds$\forall a,b \in R$iff R is commutative We want to show that for some ring$R$, the equality$a^2 - b^2 = (a-b)(a+b)$holds$\forall a,b \in R$if and only if$R$is commutative. Here's my proof --- I'm not sure if the first part stands ... 0answers 8 views ###$gr_I (R)$finite? Is$…⊕R(-1) ⨁m⨁R_1 ⨁…$necessarily an ideal When is$gr_I (R)$(I mean "associated graded ring of I") finite? When is$gr_I (M)$(M is R-module) finite? Is Exercise 2.8 from Marley's note on "GRADED RINGS AND MODULES" true? Exercise 2.8: Let R ... 0answers 13 views ### Canonical “orientification” of a manifold? Canonical complexification of a manifold? Maybe this is a silly question(i'm pretty new to both geometry and category theory) but i was wondering: 1)Consider the category of orientable smooth manifold on$\mathbb{R}$, if you forget the ... 0answers 16 views ### existence of spanning trees in complete graphs implies choice? it is known that the existence of spanning trees in arbitrary (connected) graphs implies the Axiom of Choice. I was wondering if this result still holds if we restrict ourselves to spanning trees of ... 0answers 6 views ### Notations in Oka family definition Definition. An ideal family$F$in a ring$R$with$R \in F$is said to be an Oka family (strongly Oka family) if, for$a \in R$and$I$,$A \lhd R$,$(I, a), (I:a) \in F \Rightarrow I \in F$... 0answers 5 views ### Inducing highest weight modules I have a question regarding highest-weight modules: Let be$\mathfrak{g}$a Lie algebra,$\mathfrak{b}$a Borel subalgebra,$\mathfrak{h}$a Cartan subalgebra and$U(\mathfrak{g})$its universal ... 1answer 12 views ### Geometry of$k$-forms and$k$-vectors In this question I was trying to see why$k$-forms are selected as the way to generalize vector calculus rather than$k$-vectors and a comment providing links to other questions made me end up with ... 0answers 12 views ### Looking for hints on how to prove the convergence of this iterative estimator! [duplicate] Let$X_n$be a Poisson process with mean$\lambda^*$. The following sequence estimates the parameter of the Poisson process:$ X_{n+1} = \hat{\lambda}_{n+1} + ...
Assume I have a problem such as "Prove that $\displaystyle103^{53} + 53^{103}$ is divisible by $39$." This would mean I wanted to prove that $\displaystyle103^{53} + 53^{103}\equiv0\pmod{39}$. My ...