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    Invariance of the speed of light

    Hello! Consider the law of addition of velocities for a particle moving in the x-y plane: u_x=\frac{u'_x+v}{1+u'_xv/c^2},\, u_y=\frac{u'_y}{\gamma(1+u'_xv/c^2)} In the book by Szekeres on mathematical physics on p.238 it is said that if u'=c, then it follows from the above formulae that...
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    The Structure of Galilean Space

    A Galilean transformation is defined as a transformation that preserves the structure of Galilean space, namely: 1. time intervals; 2. spatial distances between any two simultaneous events; 3. rectilinear motions. Can anyone give a short argument for the fact that only measuring the...
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    Contraction of Tensors

    Hi all! I've got a short question concerning a minor notational issue about tensor contraction I've run across recently. Let A be an antisymmetric (0,2)-tensor and S a symmetric (2,0)-tensor. Then their total contraction is zero: C_1^1C_2^2\,A \otimes S=0. As a proof one simply computes...
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    Bounded sets

    I am a bit disturbed by the following elementary observation. Let (X,d) be a metric space and \emptyset\neq A \subseteq X. (a) The diameter \delta (A) of A is defined to be \delta (A):=\sup_{(x,y) \in A^2}d(x,y), where A^2:=A \times A (b) A is called bounded if \delta (A)<\infty. Now let...
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    Not every metric comes from a norm

    Hello! It is said that not every metric comes from a norm. Consider for example a metric defined on all sequences of real numbers with the metric: d(x,y):=\displaystyle\sum_{i=1}^{\infty}\frac{1}{2^i}\frac{|x_i-y_i|}{1+|x_i-y_i|} I can't grasp how can that be. There is a proof...
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    All natural numbers

    Hello! Question: if it is asked to prove a statement A(n_1,...,n_k) for all natural numbers n_1,...,n_k, is it actually enough to check its truth by induction on just one of the counters, say n_1?
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    Subbasis help

    Hello, could you please check if the reasoning is correct. This is not a homework, just a part of an exercise in a book I'm reading at the moment. Suppose X is a set, \mathcal{B}:=\{S\subset{}X:\bigcup{}S=X\}, \\...
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    Basic lemma in topology

    Hello, there is a basic lemma in topology, saying that: Let X be a topological space, and B is a collection of open subsets of X. If every open subset of X satisfies the basis criterion with respect to B (in the sense, that every element x of an open set O is in a basis open set S, contained...
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    Empty family of sets

    Hi! I'd like to ask the following question. Does it make sense to take unions and intersections over an empty set? For instance I came across a definition of a topological space which uses just two axioms: A topology on a set X is a subset T of the power set of X, which satisfies: 1...
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    Free homotopy

    Hello, Here is a short lemma: A path-connected space X is simply-connected iff any two loops in X are free homotopic. My question is whether it is allowed to use a straight-line homotopy straight away in order to construct a free homotopy? For example, let u and v be two loops and w is a...
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    A Property of an Autonomous ODE

    Hi! I wonder how to prove that if y(t)=sin(t) solves an autonomous ODE f(y,y',...,y^(n))=0, then x(t)=cos(t) is also a solution. I mean I'm a bit distracted by the fact that all derivatives of y are present here. For example in the equation for a pendulum there are just y and y'' and a...
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    Levi Theorem

    Hello! I've got big problems with understanding abstract algebra, the way we deal with it in the seminar on Lie algebras. In just four weeks we progressed up to Levi and Malcev theorems, which are actually the culmination, the say, of classical Lie algebras theory. I didn't think, that the...
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    Symmetry of Polynomials

    Hi! Brief question: I wonder which conditions should a polynomial function of odd degree fulfill in order to be symmetric to some point in the plane. Are there such conditions?
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    Straight lines in Polar Form

    Homework Statement Problem from Arnold's "Mathematical Methods of Classical Mechanics" on page 59. Find the differential equation for the family of all straight lines in the plane in polar coordinates. Homework Equations \Phi=\displaystyle\int^{t_2}_{t_1}...
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    Prove that [itex]f(z) = |z|[/itex] is not holomorphic.

    Prove that f(z) = |z| is not holomorphic. We have f(z) = \sqrt {x^2 + y^2} + i0, so \frac {\partial{u}}{\partial{x}} = x(x^2 + y^2)^{ - \frac {1}{2}},\frac {\partial{u}}{\partial{y}} = y(x^2 + y^2)^{ - \frac {1}{2}}, \frac {\partial{v}}{\partial{x}} = \frac {\partial{v}}{\partial{y}} = 0...
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    Constant function

    Please check whether this makes sense Homework Statement If U\subset\mathbb{C} open, path-connected and f:U\longrightarrow\mathbb{C} differentiable with f'(z)=0 for all z\in{U}, then f is constant. Hypotheses: H1: U is pathconnected H2: f:U\longrightarrow\mathbb{C}, f'(z)=0, z\in{U} 2. The...
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    Implicit Function

    Hi! I've got a question about implicit functions. I have to solve a system f(x,y,z)=0 in the neighbourhood of (1,1,1). I have a problem computing the derivative of an implicit function (x,y)=g(z), whose existence is given by the implicit function theorem when applied to the given function...
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    Square Root of Complex Numbers

    Hi! I've got a question. There is a nice formula for finding square roots of arbitrary complex numbers z=a+bi: \frac{1}{\sqrt{2}}(\epsilon\sqrt{|z|+a}+i\sqrt{|z|-a}) where epsilon:=sing(b) if b≠0 or epsilon:=1 if b=0. I've just looked it up and it's nice to use it to find complex roots of...
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    Cubic equation roots

    Homework Statement Show that for negative c (a,b,c - real) equation x^3+ax^2+bx+c=0 has at least one positive root. 2. The attempt at a solution Considering the equivalent form of the equation above for large |x|: x^3(1+\frac{a}{x}+\frac{b}{x^2}+\frac{c}{x^3})=0 we can conclude that there...
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    Trajectories of motion of bugs

    Hello! I'm thinking about the following problem at the moment: Four bugs sitting at the corners of the unit square begin to chase one another with constant speed, each maintaining the course in the direction of the one pursued. Describe the trajectories of their motions. What is the law of...
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    Uniform conevergence

    Uniform convergence Hello! I've got a short question to an example. I should check the following sequence for uniform convergence on the whole of \mathbb{R}: f_n(x)=\frac{nx(7+sin(nx))}{4+n^2x^2} It says that the conevergence is nonuniform, because...
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    Expression for velocity

    Hey guys! I've got a question. How do we get this expression for the velocity: \dot\vec{r}=\dot{r}+\frac{l^2}{m^2r^2}, where l is the angular impulse of force I thought we could do it like this...
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    Point Particle in Magnetic Field

    Hey, Guys! Could you please give me some guidance for the following problem: A point particle of mass m and charge q moves with an arbitrary initial velocity \vec{v} in constant magnetic field \vec{B}. The point particle is moving under the influence of the...
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    Initial Values

    Hi, World!! Nice place here! My first post in this forum. :smile: I've got a short question for a start. If we wish to evaluate the constants for the general solution x(t)=C_1e^{-{\lambda_1}t}+C_2e^{-{\lambda_2}t} of this ODE: \ddot{x}+2{\gamma}\dot{x}+{{{\omega}_0}^2}x=0 we can choose the...