Uniqueness Definition and 246 Threads

Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison. When used in relation to humans, it is often in relation to a person's personality, or some specific characteristics of it, signalling that it is unlike the personality traits that are prevalent in that individual's culture. When the term uniqueness is used in relation to an object, it is often within the realm of product, with the term being a factor used to publicize or market the product in order to make it stand out from other products within the same category.The notion of American exceptionalism is premised on the uniqueness of the West, particularly its well-defined secularism.

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  1. J

    Uniqueness Theorem: Qualitative Example of 1st Order Linear DE

    Can someone give me a qualitative example of the uniqueness theorem of a first order linear differential equation? I have read the definition, but I am not 100% positive of what it means in regards to an initial value problem. Im confused about what a unique solution is when/if you change the...
  2. STEMucator

    Existence and Uniqueness of solutions

    Homework Statement These questions were on my midterm a while ago. I want to understand this concept fully as I'm certain these will appear on my final tomorrow and I didn't do as well as I would've liked on these questions. http://gyazo.com/205b0f7d720abbcc555a5abe64805b62 Homework...
  3. M

    The uniqueness of a magnet? Let me tell you why.

    In all of the things that we've discovered have you ever found something as amazing like a magnet? An object that can attract/repel (by using force!) on its own similar other magnets or metals. The only object present to withhold a significant amount (depending on type,size,etc...) of force...
  4. N

    Proof of Existence & Uniqueness of Rational Number y for xy = 2

    To get the following proof I followed another similar example, but I'm not sure if it's correct. Does this proof properly show existence and uniqueness? Show that if x is a nonzero rational number, then there is a unique rational number y such that xy = 2 Solution: Existence: The nonzero...
  5. E

    Why is N=4 SYM's Lagrangian Unique?

    Hello, I've seen stated in many places that N=4 SYM is a unique theory and I'm wondering why this is. In my reading, I've seen why there is a unique N=4 supermultiplet and this would fix the field content of the theory (up to one caveat I have: why couldn't an N=4 theory contain multiple N=4...
  6. S

    Uniqueness and origin of physics theorems

    After studying some of the proofs of equations, worries arises about their uniqueness. 1.Removal of integral ∫∇．Edτ = ∫(ρ/ε)dτ →∇．E = Q/ε 2.Removal of (∇X) ∇XE = ∂B/∂t = ∂(∇XA)/∂t →∇X(E+∂A/∂t)=0 →E+∂A/∂t=-∇V →E=-∇V-∂A/∂t Origin - Certain physics equations/theorems have no...
  7. J

    Proving Existence and Uniqueness of Cut C for A+C=B

    Homework Statement Show that for any two Dedekind cuts A,B, there exists a unique cut C such that A+C=B 2. The attempt at a solution In order to prove this, I need to prove the existence and uniqueness of such a cut. For the existence, I started by considering a cut for which this works...
  8. N

    Uniqueness of canonical transformations

    The following question seems to be simple enough...Anyway, I hope if someone could confirm what I am thinking. Is canonical transformation in mechanics unique? We know that given \ (q, p)\rightarrow\ (Q, P), \ [q,p] = [Q,P] = constant and Hamilton's equations of motion stay the same in the...
  9. A

    Is the Uniqueness of the Laplace Transform Affected by Function Form?

    Consider one-sided Laplace transform:$$\mathcal{L} \left \{ h(t) \right \}=\int_{0^-}^{\infty}h(t)e^{-st}dt$$ Q. Is this defined only for the functions of the form f(t)u(t)? If no, then f(t)u(t) and f(t)u(t)+g(t)u(-t-1) are two different functions with the same Laplace transform, and thus...
  10. S

    Uniqueness Theorem: Finding region

    Hello everyone! So today is was my first day of differential equations and I understood most of it until the very end. My professor started talking about partial derivatives which is Calc 3 at my university. He said Calc 3 wasn't required but was recommend for differential equations. He...
  11. E

    Dynamic Maxwell equations, uniqueness theorem, steady-state response.

    Hello, I'm trying to make a sort of "system theory approach" to dynamic Maxwell's equations for a linear, isotropic, time-invariant, spacely homogeneous medium. The frequency-domain uniqueness theorem states that the solution to an interior electromagnetic problem is unique for a lossy...
  12. J

    How Does Rudin Prove the Existence of h in the Uniqueness of n-roots?

    Hi! I need some help here, please. In Principles of Mathematical Analysis, Rudin prove the uniqueness of the n-root on the real numbers. For doing so, he prove that both $y^n < 0$ and that $y^n > 0$ lead to a contradiction. In the first part of the proof, he chooses a value $h$ such...
  13. V

    Some queries on uniqueness theorem

    Consider a solid conductor with a cavity inside. Place a charge well inside the cavity. The induced charge on the cavity wall and the compensating charge on the outer surface of the conductor will be distributed in a unique way. How does this follow from the Uniqueness Theorem of EM? David...
  14. F

    Is Potential Flow Unique in Real Life?

    there is something i can't seem to get about potential flow, when we work with potential flows we combine some simple potential flows to satisfy some boundary condition (shape of the body and potential at infinity), we get the resulting flow and we assume that that is the flow in reality...
  15. J

    Uniqueness with Laplace's Equation and Robin Boundary Condn

    Homework Statement Suppose that T(x, y) satisfies Laplace’s equation in a bounded region D and that ∂T/∂n+ λT = σ(x, y) on ∂D, where ∂D is the boundary of D, ∂T/∂n is the outward normal deriva- tive of T, σ is a given function, and λ is a constant. Prove that there is at most one solution...
  16. M

    MHB Proving Uniqueness of Fourier Coefficients for Continuous Periodic Functions

    Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$ I know this has to do with the...
  17. H

    I just have a question about Uniqueness of Limits with divergent sequences.

    Homework Statement I'm supposed to answer true or false on whether or not the sequence ((-1)^n * n) tends toward both ±∞ Homework Equations Uniqueness of Limits The Attempt at a Solution I did prove it another way, but I would think that uniqueness of limits (as a definition...
  18. D

    Showing uniqueness of complex ODE

    Homework Statement We are given (1-z)*f'(z)-3*f(z) = 0, f(0) = 2 valid on the open disk centered at 0 with radius 1 and told to prove there is a unique solution to the differential equation. The hint he gave was to find a factor that makes the left side the derivative of a product, but that...
  19. D

    Calc 4 Student, Please help me understand Existance and Uniqueness

    Ok, after going thru the proof, the only think that still eludes me is the region of definition given in the theorem itself. The rectangle where f and the partial of f with respect to y are known to be continuous. I am thinking of this three dimensionally, and I do not know if this is the...
  20. L

    Existence and uniqueness of differential solution, help?

    Ok so ill give an example, x'(t) = log(3t(x(t)-2)) is differential equation where t0 = 3 and x0 = 5 The initial value problem is x(t0) = x0. So what i'de do is plug into initial value problem to get x(3) = 5, so on a graph this plot would be at (5,3)? Then plop conditions into differential...
  21. F

    Uniqueness of Limits of Sequences

    Here is the proof provided in my textbook that I don't really understand. Suppose that x' and x'' are both limits of (xn). For each ε > 0 there must exist K' such that | xn - x' | < ε/2 for all n ≥ K', and there exists K'' such that | xn - x'' | < ε/2 for all n ≥ K''. We let K be the larger...
  22. H

    Proving Existence and Uniqueness for x in ℝ in a Quartic Equation

    Homework Statement x/√(x2+y12)-(l-x)/√((l-x)2+y22)=0 How do I prove that the above equation has a solution for x in ℝ and that the solution is unique? (y1, y2, and l are constants.) Homework Equations x√((l-x)2+y22)-(l-x)√(x2+y12)=0 x√((l-x)2+y22)+x√(x2+y12)=l√(x2+y12)...
  23. B

    MHB Prove Existence & Uniqueness for Diff. Eq. w/ Measurable Coeff. & RHS

    Dear MHB members, Suppose that $p,f$ are locally essentially bounded Lebesgue measurable functions and consider the differential equation $x'(t)=p(t)x(t)+f(t)$ almost for all $t\geq t_{0}$, and $x(t_{0})=x_{0}$. By a solution of this equation, we mean a function $x$, which is absolutely...
  24. P

    Differential Equations - Existence and Uniqueness

    I'm having trouble understanding what uniqueness is/means. When given a slope/direction field I don't know what I should be looking for if asked to determine if a given initial condition has a unique solution. Example: \textit{y' = }\frac{(x - 1)}{y} With this equation I can see that as long...
  25. G

    Uniqueness of a completion space

    Homework Statement This question arises from Chapter ONE, section 2, ex. 7 (d) and (e) of "Introduction to Topology" by Gamelin and Greene. Xbar may be regarded as the completion of a metric space X by identifying each X with the constant sequence {x,x,...}. Show that when Y is the...
  26. T

    Showing Uniqueness of Elements of a Vector Space

    Homework Statement http://img267.imageshack.us/img267/8924/screenshot20120118at121.png The Attempt at a SolutionWe have that X = A + B. To show that X is unique, let two such sums be denoted by X1 X2 such that X1 ≠ X2. We write, X1 = A + B X2 = A + B The equations imply, X1 - A - B = 0 X2...
  27. T

    Proving Uniqueness in Subspace Addition

    Homework Statement http://img854.imageshack.us/img854/5683/screenshot20120116at401.png The Attempt at a SolutionSo we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S...
  28. T

    Proving Uniqueness in Subspace Addition

    Homework Statement http://img854.imageshack.us/img854/5683/screenshot20120116at401.png The Attempt at a SolutionSo we have that A + B is a vector in S + T, where A is an element of S and B is an element of T. Suppose there is another vector A' + B' also in S + T, where A' is an element of S...
  29. J

    Calculation and Uniqueness of Smith Normal Forms

    FYI this is a homework problem which I already have the answer to but would like to clarify some points on. Homework Statement Find the Smith Normal Form of the matrix \left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right] over the ring of integers. Homework...
  30. C

    Existence and Uniqueness of System of Differential Equations

    Hi everyone, I'm not quite sure how to proceed to show existence (and perhaps uniqueness) of the following system of (first order) differential equations: \dot{x}=f(t_1,x,y,z) \dot{y}=g(t_2,x,y,z) \dot{z}=h(t_3,x,y,z) where \dot{x}=\frac{\partial x}{\partial t_1}...
  31. P

    Newton's laws and Uniqueness of Motion

    How would you show mathematically that Newton's laws, when taken as given, always yield a motion and that this motion is always unique (given initial positions/velocities) for arbitrary systems?
  32. A

    Uniqueness theorem for Laplace's equation

    Hi all. Suppose that U1 is the solution of the Laplace's equation for a given set of boundary conditions and U2 is the the solution for the same set plus one extra boundary condition. Thus U2 satisfies the Laplace's equation and the boundary conditions of the first problem, so it's a solution...
  33. A

    Proving Uniqueness of Affine Plane Containing S & Weak-Parallel to T

    Homework Statement S and T are two affine lines in \mathbb{A}^3 that are not parallel and S\cap T=\emptyset. Show there is a unique affine plane R that contains S and is weak parallel with T. The Attempt at a Solution Existence is easy, if S=p+V and T=q+W then R=p+(V+W) satisfies the...
  34. G

    Proof of Uniqueness of y for x > 0

    Homework Statement If x > 0, then there exists a unique y > 0 such that y2 = x. The attempt at a solution Proof. Let A = {y ∈ Q : y2 < x}. A is bounded above by x, so lub(A) = η exists. Suppose η2 > x, where η = lub(A). Consider (η - 1/n)2 = η2 - 2η/n +1/n2 > η2 - 2η/n. Now η2 -...
  35. K

    Uniqueness issue of direct sum decompostion of a representation?

    I'm having difficulty understanding this concept of uniqueness. What's the precise definition of it? Let say we have some direct sum decomposition, (1)Are \left( {\begin{array}{*{20}{c}} {{R_1}} & 0 \\ 0 & {{R_2}} \\ \end{array}} \right) and \left( {\begin{array}{*{20}{c}}...
  36. G

    Integrating factor for first order linear equations uniqueness theorem

    My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...
  37. R

    Existence, Uniqueness of a 1st Order Linear ODE

    Homework Statement Solve the Cauchy problem: (t2 + 1)y' + etsin(t) y = sin(t) t2 y(0) = 0 Homework Equations y'(t,y) + p(t)y = g(t,y) Integrating factor e(integral of p(t)) The Attempt at a Solution I tried finding an integrating factor, but it came out ugly. I couldn't solve the...
  38. D

    Existence and uniqueness of PDEs

    Hello, I have a PDE: 3*u_x + 2*u_y = 0, and I am interested in determining initial values such that there is a unique solution, there are multiple solutions, and there are no solutions at all. What theorem(s)/techniques would be of use to me for something like this? Regards, Dan
  39. K

    Prove the assumption of uniqueness is not necessary

    Is my proof correct? Homework Statement Show that, if there exists a number 0 for which x+0=x for all x∈R, and a number 0' for which x+0'=x for all x∈R, then 0=0'. The Attempt at a Solution Proof by contradiction: Assume, 0≠0'. Then, x+0=x and x+0'=x' Such that: x=x-0 and x=x'-0'...
  40. L

    Is the Solution to dx/dt=f(x) with x(0)=xo Unique?

    Homework Statement Show that the solution of dx/dt=f(x), x(0)=xo, f in C^1(R), is unique Homework Equations C^1(R) is the set of all functions whose first derivative is continous. F(x)=integral from xo and x (dy/f(y)) The Attempt at a Solution Assume phi1(x) and phi2(x) are both...
  41. Y

    Query regarding existence and uniqueness of SDE solutions

    Hi, Just wanted to ask a question regarding existence and uniqueness of solutions to SDEs. Say you have shown existence and uniqueness of a solution to an SDE that the process [tex ] X_{t} [/tex ]a particular process follows (by showing drift and diffusion coefficients are Lipschitz). If you...
  42. P

    Problem for Theorem of Uniqueness

    Homework Statement Check if the given initial value problem has a unique solution Homework Equations y'=y^(1/2), y(4)=0 The Attempt at a Solution f=y^(1/2) and its partial derivative 1/2(root of y) are continuous except where y<=0. We can take any rectangle R containing the...
  43. P

    Diff Equations: Theorem of Uniqueness

    Homework Statement Check if the given initial value is a unique solution. Homework Equations y'=y^(1/2), y(4)=0 The Attempt at a Solution I got y(t)=(t/2)^2 and 0 at t=4 So, we have two solutions to i.v.p.; therefore, it's not a unique solution. Is it correct?
  44. J

    How to prove: Uniqueness of solution to first order autonomous ODE

    Hello! I would like to prove the following statement: Assume f\in C^{1}(\mathbb{R}). Then the initial value problem \dot{x} = f(x),\quad x(0) = x_{0} has a unique solution, on any interval on which a solution may be defined. I haven't been able to come up with a proof myself, but would...
  45. M

    Counterexample to uniqueness of identity element?

    (Hopefully, this question falls under analysis. I was unable to match it well with any of the forums.) The proof that the identity element of a binary operation, f: X x X \rightarrow X, is unique is simple and quite convincing: for any e and e' belonging to X, e=f(e,e')=f(e',e)=e'. However...
  46. A

    Picards Existence and Uniqueness theorem doesn't prove anything

    As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the...
  47. S

    Existence and Uniqueness theorem for 1st order ODEs

    Homework Statement Consider the IVP compromising the ODE. dy/dx = sin(y) subject to the initial condition y(X) = Y Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is...
  48. S

    Uniqueness of State Transformation Matrix for Controllable Systems

    Homework Statement Two systems are given (both are completely controllable): x-dot = Ax + bu z-dot = A*z + b*u They are related by the state transformation: z=Tx prove that the transformation matrix T is unique. The Attempt at a Solution Since the systems are completely controllable, we...
  49. B

    Uniqueness and Existence Theorem

    Homework Statement for the differential equation t^2y''-2ty'+2y=0 with the general solutions y=C(t) + D(t^2) where C and D are constants. given the inital solution y(0)=1 and y'(0)=1 there are no solutions that exist. Why does this not contradict the Existence and Uniqueness Theorem...
  50. A

    Uniqueness and existence of simplified equivalent circuits

    I know this probably sounds weird, but I have a research problem that requires "random" analog circuits. Basically what this means is that I create Spice netlists by randomly adding linear and/or nonlinear components of random types with random node and parameter values. This works fine and I...
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