enter image description here Everytime I make a calculation for a system of equations it comes out in this format, how do I change the setting so that I always get a decimal?

Question Question - What is the area of the shaded part? I tried doing area ratios but I get to nothing. This is my weak subject. Could someone please help me?

1 + 3 + 3^2 + ... + 3^(n-1) = 1/2(3^n - 1) I am stuck at (3^k - 1)/2 + 3^k and I'm not sure if I am right or not.

enter image description here Image attached of the closed form of the Fibonacci Sequence.

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. Having $$L=\frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right), M=M_{x+iy} $$ Acting on $D$. I cant understand how...

Is that it is a formula that can help to calculate the Fourier transform of $$ f(x) = x^2 e^{-b x^2} e^{-c x \coth(bx)} $$ where $b>0$ and $c\in \mathbb R$. Thanks you in advance

i just need help getting started, I'm not sure what tools to use..

A triangular piece of land has one side measuring 2ft. The land is to be divided into 2 equal areas by a dividing line parallel to the given side. What is the length of the dividing line? so far this is what i know and it's in one of my quizzes. Can any of you help me answer this please?

Let $f(x):=e^{x} \int_{0}^{x} e^{-t} g(t) dt$ then I want to show that $$\lim_{x \rightarrow \pm \infty} f(x)=0$$ if $g \in C_0(\mathbb{R}).$ Does anybody know how to show this?

Can circumference and area of a circle be measured without pi?

Prove the product expression $$\left \| AB \right \|_{U\rightarrow W} \leq \left \| A \right \|_{V\rightarrow W}\left \| B \right \|_{U\rightarrow V}$$ Hint: consider $(AB)u = A(Bu)$ and apply $\left \| Av \right \|_{W}\leq \left \| A \right \|\left \| v \right \|_{V}$ twice

How to make inversion of this (n x n ) matrix? enter image description here

Let $G$ be a compact Lie group. A "matrix entry function" is one of the form "(i,j)-th entry of the matrix $\rho(g)$" for a Lie group homomorphism $\rho$ from $G$ to a matrix group. (i.e. a representation with a preferred basis) I'd really like it if someone could show me a simple proof (basical...

I am trying to answer the following question- Let $X_n\sim Geom(\frac\lambda n)$ and $T_n=\frac{X_n}n$. Show that for $n\rightarrow\infty$ the distribution function of $T_n$ converges to the DF of Z, when $Z\sim Exp(\lambda)$. (1) This brought to the thought- does this mean the limit of $T_...

I need help with this proof. I claim it is true, and I want to prove it directly using the definition of onto. Proof: Let $A,B,$ and $C$ be sets, and let f, g be functions s.t. $f:A \rightarrow B$ is onto and $g:B \rightarrow C$ is not onto. Then $\forall b \in B, \exists a \in A$ s.t. $f(a) = ...

Can an improper Riemann integral converge while its infinite series diverges?

I'm trying to prove that there are no 3 positive integers $(a,b,c)$ such that $(a,c), (b,c), (a,a-b+c), (b,a-b+c)$ form 4 pairs of legs of pythagorean triples. Or otherwise to find such a set of integers. ie Prove the system of equations: $$a^2+c^2=w^2$$ $$b^2+c^2=x^2$$ $$a^2+(a-b+c)^2=y^2$$ $...

A is an n x n matrix here. I understand the proof for A^2 being invertible given that A is invertible, but I fail to see how to incorporate the A + A^2 factor into it. What I have tried so far is a rough factoring to give: A(I_n + A) But that is where I am stuck. Any help is much appreciated!

I saw this written on a blackboard in the math department building the other day: Gas Law: $PV=nRT$ Ideal Gas Law: $(P)(V)=(n)(R)(T)$ I know the ideal gas law is something from chemistry, but I'm assuming this is meant to be some sort of joke involving math. Any ideas?

A theorem in Euclid's Elements states that the line segment between the midpoints of two sides of a triangle is parallel to the third side and is half its length. I have looked in various textbooks in geometry and have not found a generalization of this. If $0 < p < 1$, such as $2/3$ or $\sqrt{...

\int _0^1:\frac{\left(5x\right)}{\left(5x^2+6\right)^2}d not sure how to solve this problem

Prove the product expression $$\left \| AB \right \|_{U\rightarrow W} \leq \left \| A \right \|_{V\rightarrow W}\left \| B \right \|_{U\rightarrow V}$$ Hint: consider $(AB)u = A(Bu)$ and apply $\left \| Av \right \|_{W}\leq \left \| A \right \|\left \| v \right \|_{V}$ twice Unfortunately, I'm...

Let $A\left( {\begin{array}{*{20}{c}} {{x_1}} & 0 \\ 0 & {{x_2}} \\ \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ \end{array}} \right) = \lambda \left( {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ \end{array}} \right)$ where $A \in {M_2}(R)$ and all $a_{...

I want to know if it is possible to calculate the number of inputs to any operator in Reverse Polish Notation. Suppose I have an operator ^ that takes varying amounts of inputs, specific to each instance of the operator. The input count is not known. All other operators are removed. In this examp...

Given a family of curves $\mathcal{F}_1$ and an orthogonal family of curves $\mathcal{F_2}$, how would one move from these families to an orthogonal curvilinear basis? I have trouble going from the equation representation of the curve families to a vector/matrix representation.

Let $m, n, k$ be natural numbers, such that $$m^3=n^2$$ $$m+n=k^2$$ What is the largest possible value for $m$ if $m < 1000$

Let $\mathscr{A}$ be a linear transformation on $n$-dimensional vector space $V$. $0\neq\alpha\in V$. Then it is easy to see that there exists a unique polynomial $m_\alpha(\lambda)$ such that $m_\alpha(\mathscr{A})(\alpha)=0$, and for any polynomial $f(\lambda)$ satisfying $f(\mathscr{A})(\alph...

$\lim_{x\to \infty} \frac{x^2(1+sin^2x)}{(x+sinx)^2}$ I can't figure out how to manipulate this algera so as to get the limit I want. Any hint?

Find the volume of solid generated by region in the first quadrant bounded by curve $y=x^2$, below by the x axis and on the right by the line $x=2$ and about the line $x=-3$ How to set up integral in this question? Please help.

I would like to evaluate $$ \prod_{i=1}^{N} \sum_{q_i = -\infty}^{\infty} e^{- q_{i} A_{ij} q_{j}} $$ with $A$ a real $N\times N$ symmetric matrix. I know how to compute this when $q$ is continuous (the sum is an integral), and I know how to compute this when $A$ is a scalar (a $1\times 1$, this...

So I've been trying to find the volume enclosed by this area for hours and when I do the triple integral with φ from 0 - π, θ from 0 - 2π and p from 0 to 2cos(φ)sin(φ)^2 I get 0!!! QQ Please help me out I'm stumped Here is a graph of the equation

I have a problem with solving these 3 symmetric inequations. I want to find positive values for these 6 variables such that the following inequations hold (or show that it is impossible). Please suggest me how to do, or any good tools for this: $\frac{2a}{b+c} \lt \frac{b'+c'}{a'}$ (1) $\frac{2...

The system is this: $\frac{dx}{dt}=4x-7y-1$ $\frac{dy}{dt}=3x+6y-12$ Okay i found the equilibrium points to $(0,0)$ and $(2,0)$ but I also read somewhere else that the only way to find the equilibrium points to be trivial is that the det $A$ $\not=0$. This leads me to the phase portrait and di...

This question just occurs to my mind... It is closed clearly from the graph, but I'm wondering if there is a rigorous proof for it.

Consider the alternating series: sin (x) = x - x^3 / 3! + x^5 / 5! .... If we approximate sin(x) ≈ x, then |x - sin (x) | < | answer | What would be the answer, I tried various methods, but all of them led to wrong answers. Please do explain the working.

In $$ L^{2} (\mathbb{R}^2, e^{{-x^2}-y^{2}} dx dy)$$ with subspace $D$ of finite linear combinations of $g_m=(x+iy)^m$ , $m\neq 0$ and integer $(g_0=1)$. Having $$L=\frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right), M=M_{x+iy} $$ Acting on $D$. I cant understand how...

Let $Y_i$ be i.i.d. with distribution $P(Y_i = 1) = p$, $P(Y_i = 0) = 1-p$, $0<p<1$. Let $N_t$ be a Poisson Process (with parameter $\lambda$) and let $T_i$ be the arrival of the $i^{th}$ event. Define $$V_t = \sum_{i=1}^{\infty} (1-Y_i) 1_{T_i \leq t} $$ Show that $V_t$ is a Poisson Proc...

I think this is a problem closely related to dynamical system... Let $T(t)$ be a $C^{0}$ semigroup of contractions on a Banach space $X$, and assume that the resolvent $R(\lambda,A)$ of the infinitesimal generator $A$ of the semigroup is a compact operator for some $\lambda>0$. I need to prove ...

Let $a$>0 be a fixed positive real number. Consider the solid inside both the cylinder $x^2 + y^2 = ax$ and the sphere $x^2 + y^2 +z^2 = a^2$. Compute its volume. Hint: $$\iint sin^3(x)dx = 1/12cos(3x) - 3/4cos(x) + C$$ Please help I have my exam soon and I don't understand anything about cylind...

In a tennis tournament in which every pair has to play a match with every other pair, $10$ players are playing.Find the number of games played. ATTEMPT:-No of ways of selecting $2$ players out of $12$ for forming a pair is same as arranging them in a row and then dividing them into groups of t...

Here y^2 is divisible by 12. And satisfying all those conditions y=0 is the only solution. But I can't show it mathematically.

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, $\alpha^\beta$ is the least ordinal greater than any ordinal in the set $\{\alpha^\gamma:\gamma<\b...

There are 4 different Mathematics books and 5 different Science books. In how many ways can the books be arranged on a shelf if there are no restrictions?

I'm studying for finals and have encountered an issue with a problem. I would greatly appreciate help in finding where I screwed up. $$\int \frac {cos27t^{1/2}dt}{(27t)^{1/2}}$$ I changed it to $$\int (cos27t^{1/2})(27t)^{-1/2})dt$$ and set $$u=27t^{1/2}$$ with $$\frac {du}{dt}=\frac{1}{2}(27t)^{...

Find the derivative of tan(radical(1-x)) So I know I have to apply the product rule so wouldn't it be sec^x(rad(1-x))+tanx/2(rad(1-x)) but the final answer says -sec^2(rad(1-x))/2(rad(1-x))

divergence question Above is the question. I've try to find the divergence of F and parameterize the sphere using spherical coordinates. Below is my work. Then I use online integral calculator(just to avoid human error) to find the result is 100000pi/3, but the result isn't right. Is anything wr...

In Bochner's thoerem, one condition is $\phi$ is positive definite, which is $\sum_{j=1}^{n}{\sum_{k=1}^{n}{\phi(t_j-t_k)z_j\bar{z_k}}}\geq 0$ for all $z_j\in\mathbb{C}$ and $t_j\in\mathbb{R}$. My question It is the definition of nonnegative definite, not positive definite.

Evaluate ∬F.ds where F=xi+yj+4zk and S is the surface of the volume: V={(x,y,z)|x^2+y^2≤9,0≤z≤1} Find divF ∬F.ds = ∫∫∫ divF dxdydz = ?? Please explain in detail how to get the answer.

Let square matrix $A$ of size $n \times n$, have entries that have been independently sampled from a uniform distribution between $[a_1,a_2]$. The question I have is what distribution will the eigenvalues fall under? My Approach: My exploration began by looking at the characteristic polynomial ...

Find the revenue and demand functions for the marginal revenue $\frac{dr}{dx}=50+7x-37x^2$ How can I solve this problem?

Fibonacci Numbers fn are defined recursively as fn = fn-1 + fn-2 for n > 2 and f1 = f2 = 1. They also admit a simple closed form: √5𝑓𝑛 = (1+√5/2)^2 - (1-√5/2)^2 Prove this formula by using induction.

Let $K$ be a compact subset of $\mathbb{C}$. Then the polynomially convex hull $\hat{K}$ and the convex hull $co(K)$ of $K$ are defined as follows: $\hat{K}=\{z\in \mathbb{C}:|p(z)|\leq max_{\zeta \in K} |p(\zeta)|$ for all polynomials $p\}$. $co(K)=\{z\in \mathbb{C}:L(z)\leq max_{\...

Find y': yx^3=4e^(xy) how do I go about finding this derivative? Step by step explanation please!

Does its shape have a purpose? If so, what is it?

I'm trying to determine the quadrature formula when the interval is [−2, 2] and the nodes are −1, 0, and 1. However, I get stuck on this step because I don't understand how the attached image is equal to $1/2x(x-1)$ My attempt is, $(x - 1)/(-1 - 1)$ and $(x - 2)/(-1-2)$ but that is completely...

Can anybody please tell me whether there are any self complementary trees and if so, how many and what are the conditions and the properties it holds? Thank You.

If you are given a sum, say $$\sum_a^b f(x)$$ with $a,b\lt \intfty$ Is there a means of solving for this sum by means of integration? I am familiar with sophomores dream.

$$\frac{dx}{dt}=4x-7y-1$$ $$\frac{dy}{dt}=3x+6y-12$$ one part of my notes says if the determinant gives you some number other than zero then there is only one equilibrium solution (the origin). However another part of my notes says that the equilibrium solutions are just setting each equatio...

I want to show that the function $f(x)=(-1)^x$ is such a function. Is it sufficient to show that the limit of two subsequences of $x$ are not equal (i.e. $1$, $-1$)?

In the Van Dalen's Logic and Structure how is cardinality of a language defined? what about models?

I am reading Weibel's book about homological algebra. In section 6.5, the bar resolution, he uses some notation I really do not understand. So given G, a group, what is $[g_1\otimes...\otimes g_n]$? And what is $[g_1|...|g_n]$?

There is incorrect statement If $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is continuous in each variable separately, then it is continuous. Counterexample is $$\frac{2xy}{x^{2}+y^{2}}$$ as it is discontinuous at $(0,0).$ Now i am searching the condition under which the above stated incorrect st...

I am wondering whether flat module is a adapted class to the functor F(M)=M$\otimes$ N. Currently I am stuck by showing every module is a quotient of flat-module. This is too bizarre for me, I am not sure whether this is true. Could any help me?

How do you prove $\lim_{x\to\infty} \frac {1}{n^{1/n}}$ using only basic limit theorems? I thought it was $0$, but my book lists the solution as $1$. How come?

I am am very confused about a fundamental result in representation theory of finite groups. Please let me first introduce the setting. Let $G$ be a finite group. The group algebra $\mathbb{C}[G]$ is a semisimple ring by Maschke's theorem. Hence, there is an isomorphism $$ \mathbb{C}[G]\cong U_1\...

Suppose that fn and f are measurable functions such that for each e> 0 we have ∑μ({x:|fn(x)-f(x)|>e})<1. Prove that fn -> f a.e.

Define $n=(x + iy)/\sqrt{2}L$ and $\overline n=(x - iy)/\sqrt{2}L$. Also, $\partial_n$ = $L(\partial_x - i \partial_y)/\sqrt{2}$ and $\partial_\overline n$ = $L(\partial_x + i \partial_y)/\sqrt{2}$. with $\partial_n=\partial/\partial n$, $\partial_x=\partial/\partial x$, $\partial_y=\partial/...

I start the proof off but saying we must show that $f:X\to$X must be injective, surjective, and bijective.

Let S be the union of two surfaces, $S_1$ and $S_2$, where $S_1$ is the set of $(x,y,z)$ with $x^2+y^2 =1$ ,$0<= z <=1$ and $S_2$ is the set of $(x,y,z)$ with $x^2+y^2+(z-1)^2 =1$, $z>=1$. Set $F(x,y,z)$=$(zx+z^2y+x)i + (z^3yx+y) +(z^4x^2)k$ Compute $\int\int_S curlF *dS$ (*: inner product) S...

My teach told me there are two equations.

I need help with these infinite series! Having a hard time figuring out where to start:

When I run this in WolframAlpha, it directly subtracts $3$ from a decimal approximation of $\pi$, giving $$0.1415926535897932384626433832795028841971693993751058...$$ However, in pure mathematical terms, what is $\pi-3$? Does a closed form exist for it (excluding '$\pi-3$')? Is it irrational, an...

Please I want to know what kind of trigonometric basis I can use instead of polynomial basis for construction of numerical scheme.

Is it enough to show that no single element of $\mathbb{Z}_7$ squared is equal to -1? Or is there more to it than that?

I am confused with the notation $\mathbb{Q}[x,y]/(x)$. I understand that $\mathbb{Q}[x,y]$ is the ring of polynomials in two variables over $\mathbb{Q}$. So $(x)$ has to be an ideal. Could anyone explain what $(x)$ means? Thank you

I have a math from my teacher and I can't find the answer, please help me: - Tom has to deliver oranges from A to B with distance (d - Ex: d=1000) and total oranges (n - Ex: n=3000) and vehicle capacity (m - Ex: m=1000). And vehicle will loses 1 orange every 1 unit of distance, the question is fi...

So my question is suppose $\xi =\exp{\frac{2 \pi i}{7}}$, then does the polynomial $x^7-2$ reduce in the field $\mathbb{Q}[\xi]$? If so how?

Let {W_t,t larger or equal 0} be a standard Brownian motion and X_t=mut+sigmaWt.Where mu and sigma are constants, and sigma greater than0. 1.What is the mean of vector=(X1,X2)? 2.What is the covariance of this vector? 3.What is the joint distribution of this vector? 4.Let a b c be constant.W...

How to calculate the controls of this curve if I know three points: start, one on the curve and the end? Here is the curve with the coordinates I know: The curve with the points I've never done this kind of calculation before, so please be patient and explain it.

Let $F$ be a field and $r$ and $s$ positive integers. Prove that, in $F[x]$, $GCD(x^r-1,x^s-1)=x^{GCD(r,s)}-1$. If $r$ and $s$ were known numbers, I could be able to attempt the problem, but I don't know how to do about this.

The sum of the first two terms of a geometric sequence is 1100, and the sum of the infinite sequence is 3600. Find the common ratio given that r is positive.

R is an injective R-module . For every two ideals I and J we will have Ann(I∩J)=Ann(I)+Ann(J). I made an effort for an hour but it was not helpful.

I'm looking for the first 10 smallest non-zero eigenvalue in a symmetrical, sparse, hermitian Matrix A (3000x3000). There are also a lot of zero-eigenvalues in the eigenalue-spectrum which are not of interest. Matlab's eigs lets me define a sigma (starting point), but the result always gives me...

If f''(x) is continuous, then does this mean that f'(x) is continuous and f(x) is continuous

I was playing a game on my phone when a question pop up on my screen coming from one of my best mathematics masters: If we know that all of the matches are in the same size, what would be the alpha's degree?

Prove that $\int_0^1 f(x)dx=0$ if $f(\frac{1}{n})=1$ for $n=1,2,3,\ldots$ and $f(x)=0$ for all other $x$. Lemma: If $g:[a,b]\rightarrow \mathbb{R}$ is a function such that $f(x)=\mathbb{1}_{\{c\}}$ for some $a<c<b$, then $\int_a^b g(x)dx=0$. Proof of the lemma: Consider a partition of ...

$A_1$ and $A_2$ are two circles in a plane. The common external tangent to $A_1$ and $A_2$ consists of length $2017$. The common internal tangent consists of length $2009$. Find $r_1*r_2$ the product of the radii. This is fairly complicated. The solution uses $(r_1 - r_2)^2 + 2017^2$, but ...

The question is question description And the answer isThe only state with period> 1 is 1, which has period 3. I don't understand why other states like 2,3,5,6 are not with period 3, they can also take 3 steps back to themselves, can't them?

I was recently given this problem: Suppose we have a triangle ABC and let there be a cevian AD. Let the foot of the perpendicular from B to AD be P and let the intersection of the extension of BP to AC be Q. Prove that DQ and AB are parallel. So I extended a line from D parallel to AC and let it...

prove that $K(G) \leq i(G)$ where $K(G)$ is dominating number of $G$ and $i(G)$ is indepent dominating number.

I am doing some exercises on groups and I am having trouble trying to show why the answer to: 5x+1 = 13 mod 23 [where the equals sign means congruent] has the answer: 18 + 26Z [where Z is the set of integers] can anyone please explain? Thanks for you help.

I have been doing some exercise on graphs and I have been working on it for days and I don't even understand the solution and the proofs. For question 44 to 47, can anyone lend me a hand? The text solution that I am having is very confusion and i can't visualize or wrap my head around it.. Thanks...

I am trying to write an exam paper for my students on computer but i am trying to find a mathjax editor that i can write them and print them.I tryed to search on internet but i did not find anything,so any suggestion would be appreciated.

Im trying to solve this exercise, but I can't... Let S be a multiplicatively closed subset of A.M a A-module,N a submodule of M and $\phi_M:M\longrightarrow S^{-1}M$ homomorphism given by $\phi_M(m)=\frac{m}{1}$. Therefore, the following are equivalent: There exists a $S^{-1}A$-submodule P of...

$c_{n} = \frac{1+(-1)^{n}}{2}$ $S_{n} = c_{1} + c_{2} + c_{3} + ... + c_{n}$ Proove that $lim \frac{S_{n}}{n} = \frac{1}{2}$ These are my steps $\rightarrow S_{n} = \frac{n}{2n}(\frac{1+(-1)^{n}}{2}) = \frac{1+(-1)^{n}}{4}$ $\frac{1+(-1^{n})}{2}$ is $2$ or $0$, so the lim of the sum is $\fr...

In $$2^{x+1}-2 \equiv 0 \pmod{29} \quad and \quad 2^{x+1}-4 \equiv 0 \pmod{28}$$ How can I find for which $x$ this holds true. $x$ is a positive integer

Please prove that If function f is concave, g:R->R is decreasing, then g*f (composition) is convex

How to solve the following equation? y''-y=-2cosx; y(0)=1, y'(0)=0; I have to find a solution to the Cauchy problem, but I can not:(

Let $v_1,\cdots,v_n$ and $w_1,\cdots w_n$ to be two equivalent vectors groups in $\mathbb{R}^m$, that is, $v_i$ can be linearly expressed by $w_1,\cdots w_n$, and $w_j$ can be linearly expressed by $v_1,\cdots,v_n$. Can we show that there exists an invertible matrix $P$ such that $$(w_1,\cdots w_...

The Big and the Small Empire are both rectangular islands and divided into rectangular landscape. In each province there is a road that runs along one of the diagonals. On each island exist roads that make a closed route, which does not go through any point several times. The picture shows the Li...

Find the probability that the random placement of n balls in m boxes, there is exactly s boxes containing exactly k balls. (assume that, s*k <= n).

Is  $\lim_{x\to 0} x+x^2=(\lim_{x\to 0} x)+x^2=0+x^2=x^2$ or $\lim_{x\to 0} x+x^2=\lim_{x\to 0} (x+x^2)=0+0^2=0$?

Hi Can anyone help on this? Consider the set which consists of all sets whose elements are natural numbers. I need to define an infinite chain and anti chain, on this set, where the ordering is by inclusion. So I did: Anti chain - the set of prime numbers, since prime numbers are a subset of ...

At the moment I try to understand the topic "proving compact sets". 2 examples: I want to ask, if my assumptions/conclusions are right. Example 1: $(x1-1)^3 + x2 \le 0,x2\ge0$ This set closed because there are just greater/less than equal signs. This set is not bounded, because if x1 goes to ...

How to solve this limit without L'hopital rule and Taylor series?

Can you please identify what class of problem this is so that I can research algorithms for solving it please? Its a a set of linear equations and inequalities/constraints looking like this: aq = 3*ct bq = 1*dt + 2 cq = 1 dq = 2 at = if (aq >= bq) then 1 else 0 bt = if (aq < bq) then 1 else 0 c...

Compute $\lim_{x\to 0} \frac{x-sinx}{x-tanx}$ with L'hopital rule. Well, I know that I need to use the Quotient rule first $(\frac{f}{g})'$. but then, what do I have to do next?

Can anyone help me on this? Consider the divisibility partial order on A, so a ≤ b if b = ma for some integer m. Is this a total order? I need to write a maximum chain and maximum anti chain if they exist. I Know that to be a a partial order, it need to obey: If a ≤ b and b ≤ a then a = b ...

Consider$$B := \left\{u \in C^2([0, 1]) : \sum_{i=0}^2 \sup_{x \in [0, 1]} \left|u^{(i)}(x)\right| \le 1\right\}$$as a subset of $C^1([0, 1])$. How do I see that it is compact in $C^1([0, 1])$?

I use strong induction on $p$. Proof. We want to show that $\forall q\in \mathbb{N} \big[q>0 \rightarrow \neg\exists p\in\mathbb{N}\big(p/q=\sqrt{2}\big)\big]$. Let $q$ be arbitrary natural number and $q>0$. Inductive Hypothesis. Let $k\in\mathbb{N}$ and $k<p$ such that $\big[q>0 \rightarrow \n...

a) Find det(B) in terms of k; b) For what value(s) of k are the column vectors of B linearly dependent; enter image description here

I have to proof: $F$ bounded in the ball $\rightarrow$ $F$ continuous Suppose (absurde) exists a sequence $\{ x_n \}$ with $\left \| x_n \right \|=1$ $\forall n \in N$ such that $\frac {F(x_n)}{\left \| F(x_n) \right \|} \xrightarrow{n \to \infty} + \infty$. So: $$\frac{x_n}{F(x_n)}= \frac{x_n}...

Let ABC be a triangle. Let Γ be it’s circumcircle, and let I be it’s incenter. Let the internal angle bisectors of ∠A,∠B,∠C meet Γ in A',B',C' respectively. Let B'C' intersect AA' at P, and AC in Q. Let BB' intersect AC in R. Suppose the quadrilateral PIRQ is a kite; that is, IP = IR and QP = QR....

In how many ways can be get a sum greater than x for n distinct numbers where each number can be between 1 and (x-1), both inclusive? Foe example- For x = 5 and n = 3, the required combinations are (1,2,3), (1,2,4), (1,3,4) and (2,3,4) i.e. total 4 ways. Similarly for x = 6 and n = 3, the requir...

Taking into account the Shannon entropy, I was wondering that, if we have a String like 1122344444455 , is this possible to find out the entropy of digit 4 in this String? In other words, I would like to know if we can find a way to measure the degree of uncertainty of occurrence of digit 4 in th...

$0 \leq a\leq \frac{1}{4}$ $a_{1} = a,$ $a_{n+1} = a + (a_{n})^{2}$ I have to proove that the sequence has a limit, and to find the limit $a_{n}$. Actually, I have no idea what to do. I know that $a_{2} > a_{1}$ and for $n>2 ,$ $a_{2} > a_{n}$ But what's next ? Thanks.

Show that there are infinitely many triples (x,y,z) of integers, such that x^3 + y^4 = z^31.

Ho do I go about solving such type of problems? $gcd(a,b) * [a.b] = ab$ where $[]$ represents greatest integer function. Just want a direction to start..

Given problem is $J[y] =\int_{0}^{x_1}y'^2dx$ with $y(0)=0$ and $y(x_1)=-x_1-1$. After solving Euler Lagrange equation I got $y=Ax+B$ . And using first boundry conditon I got $y=Ax$ We have transversatity condition $[F+(\phi'-y')F_{y'}]=0 $ at $x=x_1$. But solving this I am getting $A=0$ ...

Matrix is upper triangular and for each $k = 1, 2, ..., n$ we got: $rank(A - k \cdot I_{n}) < n$. Find the sum of the elements on the main diagonal.

I need to calculate limit $$‎\lim‎_{ ‎r\rightarrow ‎\infty‎}‎‎\frac{\Gamma(r\alpha)}{\Gamma((r+1)\alpha)}‎‎$$ where $0<\alpha <1$ and $\Gamma(.)$ is Gamma function. with thanks in advance.

f:R^n->R prove that the two statements are equal 1) For all x,y in R^n, and for all t in [0,1], f(tx+(1-t)y)>=min{f(x),f(y)} 2) For all k in R, {x : f(x)>=k} is a convex set.

Let M and N be linear subspaces of a Hilbert space $H$ with $M \perp N$. Show that $M^{\perp\perp}\perp N^{\perp\perp}$. Is it true that $M^{\perp}\perp N^{\perp}$, or $M^{\perp\perp\perp}\perp N^{\perp\perp\perp}$ ?

Is it possible to find such a function satisfying: The function f is Riemann integrable on [0,1]; For ANY interval in [0,1], there are always both positive and negative values of $f(x)$. I didn't find any way to disprove such function but did't find a concrete example either.

Prove that any continuous function on a closed interval in $\mathbb{R}$ is integrable. Let $f:[a,b]\rightarrow \mathbb{R}$ be a continuous function. We want to show that for any $\epsilon>0$ there is $\delta>0$ such that whenever $S_1$ and $S_2$ are Riemann sums corresponding to partitions...

Let $A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?

Prove that three statements are equal. 1) f(x) is a concave function. 2) For all x, x*, f(x)= 3) For all x, D^2f(x)=H (Hessian) is negative semi-definite.

So I'm having trouble finding a pattern when dealing with these types of questions; I need to find a better way to solve them: Here's the one i'm currently dealing with: Find the range space and rank of the map: a) f: R2 → P3 given by (x,y) --> (0, x-y, 3y) (these are vectors btw) So I ...

How do you solve this question: Consider two independent random variables X and Y such that X ~ B(2, a) and Y ~ B(2, b). Let W be the random variable that represents the product of each value of X with each value of Y.Construct a table showing the probability distribution of W. Hence find an exp...

I am trying to solve the following ODE: $$y''(x)+\left( k_0-\frac{\lambda}{1+cosh^2(ax)}\right)y(x)=0 \, \, \,\,\, k_0,\lambda,a>0$$ when as $x \rightarrow \infty$ the solution is of the form $y(x)=e^{ik_0x}$. My attempt: I did the folllowing substitution $t=-cosh^2(ax)$ and the ODE which cam...

32f(1) = -3f(-4) + 10f(-2) + 30f(2) + 5f(4) solve using Lagrange interpolation formula.i have tried to solve this question but still no solution found.if any one have solution regarding this question then please share it.

Why can the first term of the Taylor series expansion of $cos(\theta)$ be written as $cos(\theta_0 - (\theta - \theta_0) sin (\theta_0))$?

Let $S$ be a collection of subsets of $\{1,2,...,100\}$ such that any two sets in $S$ has non-empty intersection . Then what is the maximum possible cardinality of $S$ ?

Dears, I had prepared a noisy image. I need to compare the original image w.r.t. noisy image. For this I found Pearson Correlation coefficient and SSIM (Structural Similarity Index). I wonder the difference between these two and which is best for Image comparison. Is there some other method to c...

Prove that for all primes p: $φ(p^i)$= $p^i$ - $p^{i-1}$ I found a proof on the wikipedia article of the Euler's totient function. But I cannot understand it, as it's been many years since I dealed with math and proves. Is there maybe a longer explanation somewhere, or can you explain it in deta...

Ten different letters are given. Five letter words are formed from these given letters. The number of words having at least two letters repeated is ? Please explain your solution

I know that every smooth manifold admits a triangulation. Does this mean that it also admits a smooth $\Delta$-complex structure?

Hi can anyone help me on this problem? I have the set {0,1,3,8,9} and I want to define an example of an equivalence relation. I know that to be a equivalence relation it needs to be Reflexive, symmetric and transitive I also now that for a set of 5 elements there are 2^n^2, so if n=5 there ...

Let V be the vector space of all polynomial functions from R[x]<3 to R[x]<3 . Consider the linear forms f_i defined for p in V, as f_i(p)=p(a_i), where a_i is in R, for i in {1,2,3,4}. a) Determine under what conditions {f1,f2,f3,f4} is a basis for V*. b) Suppose that you have found the condition...

We have a 3 labelled colours, blue, green, red. n of the labels are to be arranged in a line so that no two consecutive labels are both red. Let $H(n)$ be the number of ways this can be done. Explain why H(n) satisifies the following: $H(1)= 3, H(2) = 8, H(n)=2H(n-1) + 2H(n-2), \forall n \ge...

Let V and W be finite dimensional vector spaces over R and let T_1 :V--->V and T_2:W--->W be two L.T's whose minimal polynomials are given by f(x)=x^3+x^2+x+1 and g(x)=x^4-x^2-2 let T :V+W--->V+W be LT defined by T(v,w)=(T_1(v) ,T_2(w)) for (v,w) in V+W and let h(x) be the polynomial of T. Wha...

f:R n →R f:Rn→R prove that the two statements are equal 1) For all x,y x,y in R n Rn , and for all t in [0,1] [0,1] , f(tx+(1−t)y)>=minf(x),f(y) f(tx+(1−t)y)>=minf(x),f(y) 2) For all k k in R R , x:f(x)>=k x:f(x)>=k is a convex set.

Let $t_n\ge 0$. How to prove that $\lim\limits_{n\to \infty} \sup t_n=\lim\limits_{n\to \infty} \sup t_n\sqrt[n]{n}$? Here is my sketch: So $t_n\ge 0$ and $\sqrt[n]{n}\ge 1$ then $t_n\sqrt[n]{n}\ge t_n$ hence $$\lim\limits_{n\to \infty} \sup t_n\sqrt[n]{n}\ge\lim\limits_{n\to \infty} \sup t_n.$$...

Find the cumulative distribution function of X = U1/U2. I'm not really sure where to go from here. I've seen an example that uses the expected values and I'm not sure why.

Let $A$ be a subset of $[0,1]$ with non-empty interior ; then is it true that $\mathbb Q+A=\mathbb R$ ?

I'm trying to solve an integral equation (on the screenshot) as a part of finding an answer to this question but Maple says "Warning, solutions may have been lost".

I would like to ask how to evaluate equation 7? I have spent hours and still have no idea how to get a(k).

We have two circles (with optional radii) and one point outside of these 2 circles(optional position). how can we draw a circle or circles (locus' circles) so that pass the point and be tangent with two circles? (need a basic solution for finding all locuses)

Prove that the serie: $$ \sum _{n=1} ^\infty \frac{(-1)^{\lfloor n/3 \rfloor}}{n} $$ I tryed to use Leibniz rule but I did not succeed. Any suggestions? Thanks for helpers!

I was asked to prove following statement: $\log_a{(x_1.x_2)} = \log_a{(x_1)} + \log_a{(x_2)}$ What I did was: $\log_a{(x_1.x_2)} = \log_a{(x_1)} - (-1).\log_a{(x_2)}$ $\log_a{(x_1.x_2)} = \log_a{(x_1)} - \log_a{(\frac{1}{x_2})}$ $\log_a{(x_1.x_2)} = \log_a{(\frac{x_1}{\frac{1}{x_2}})}$ $\lo...

I really do not know where to start with this question

Can the following summation be written in a finite number of terms or as an integral: $$/sum_{r=1}^{/infty}/frac{/tan(/theta/2^n)}{2^(r-1)/cos(/theta/2^(r-1))}$ I tried to simplify the expression using trigonometric identities and then converting the infinite summation into a definite integral...

I don't know how to prove that the following equation is correct: Equation to prove

I am not good in computer programming at all, but I know that it will take a lot of times to perform primality test on huge numbers (10 million or billion of digits). But I particularly get interested on fermat numbers,which is numbers of the form $2^{2^{n}}$$+$$1$. And the smallest fermat number...

I'm a math major. I like math. I'm comfortable with it. I'm considering to do a PhD. The thing is I always fall behind. I can't keep up with the professor, can't turn in satisfactory homework in time, can't do well at tests either. When I learn something, I want to understand it. I want it...

If Machine accepts the string 'W' then 'W' is the part of M other wise not. Prove that membership problem is undecidable

prove or disprove: if E in R is a Null set, than its closer is also a null set. I think its true. I have no sense of direction on how to start.. thanks in advance for any response..

$$2cosx/(1+cos2x)=(1-cos2x)/sin2x$$ Here's what i did $$2cosx/(1+cos^2x-sin^2x)=(1-(cos^2x-sin^2x))/2sincosx$$ $$2cosx/2cos^x=(sin^2x+sin^2x)/2sinxcosx.$$ How do i continue?

You are given a transfer function G(s)=1.81K(s+20)/(s3+10s2+32s+32). This system is connnected with unity negative feedback. I've tried so many things but I can't do it :( I've did 1.81K(s+20)=0, but it's clearly wrong. I've got the zeros for the bottom part (s=-2 or s=-4(2x)). So, 1.81K(s+20)/s...

I have come across a question: Define $f:M(F) \rightarrow F$, by $f(A)=tr(A)$. $M$ is n by n. For the first part, I have proven it was a linear functional. Done. Now, show $f$ is onto. Is $f$ one-to-one? What am I looking to do? I don't know what I'm even looking to prove. Walk me through it ...

Prove that $\frac{n-b}{n}*\frac{n-b-1}{n-1}*\frac{n-b-2}{n-2}*...*\frac{n-b-(k-2)}{n-(k-2)}*\frac{b}{n-(k-1)}=\frac{(n-b)!}{(n-b-k+1)!}*\frac{(n-k)!*b}{n!}$

$x^{3} \geq 25$ and x is rational. I have find what is the inifimum and minimum of x. I know that there is no infimum because x is rational but i have to proove it somehow. I know there is some way to proove it with the rule that for every real number there is bigger natural number. Can you hel...

Suppose that $v_1 \neq v_2 \neq ... \neq v_n$ are eigenvalues of a matrix $A$, $n>3$. We know that eigenvectors form a subspace of $R^n$. But is it true to say that, if we take a subset of these, for example $\{v_1,v_2,v_3\}$, those span a subspace of $R^n$ of dimension $3$?

Complet the square $-3x^2+9y^2+6xy-4y-1$

I have the following integral: $F(t) = \int_0^tF(y)dy$ How can I convert that integral into differential equation and solve it?

i have a problem with understanding manifolds. Definition is quite unclear for me. Would be more than glad for intuitive explanation. And in addition i need to find whether $$M=\{(x,y,z): x^2y^2+y^2z^2+z^2x^2=xyz; x,y,z>0\}$$ is manifold or not. $F(x,y,z)=x^2y^2+y^2z^2+z^2x^2-xyz$ and $M=\{(x,y,z...

I have been stuck with this problem for quite large time. https://www.hackerearth.com/code-monk-bit-manipulation/algorithm/when-the-integers-got-upset/. In short what is says is: There are two arrays A and P of length N. There is a third array Z whose values are calculated as follows: Z[i]=(A[i]...