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Uniqueness Theorem
In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: * Alexandrov's uniqueness theorem of three-dimensional polyhedra * Black hole uniqueness theorem * Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. * Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients. * Division theorem, the uniqueness of quotient and remainder under Euclidean division. * Fundamental theorem of arithmetic, the uniqueness of prime factorization. * Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. * Picard–Lind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetism Uniqueness Theorem
The electromagnetism uniqueness theorem states the uniqueness (but not necessarily the existence) of a solution to Maxwell's equations, if the boundary conditions provided satisfy the following requirements: # At t=0, the initial values of all fields (, , and ) everywhere (in the entire volume considered) is specified; # For all times (of consideration), the component of either the electric field or the magnetic field tangential to the boundary surface (\hat n \times \mathbf or \hat n \times \mathbf, where \hat n is the normal vector at a point on the boundary surface) is specified. Note that this theorem must not be misunderstood as that providing boundary conditions (or the field solution itself) uniquely fixes a source distribution, when the source distribution is outside of the volume specified in the initial condition. One example is that the field outside a uniformly charged sphere may also be produced by a point charge placed at the center of the sphere instead, i.e. t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard–Lindelöf Theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Theorem Let D \subseteq \R \times \R^nbe a closed rectangle with (t_0, y_0) \in D. Let f: D \to \R^n be a function that is continuous in t and Lipschitz continuous in y. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval _0-\varepsilon, t_0+\varepsilon/math>. Note that D is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within D. Proof sketch The proof relies on transforming the differential equation, and applying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Quantification
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols " ∃!" or "∃=1". For example, the formal statement : \exists! n \in \mathbb\,(n - 2 = 4) may be read as "there is exactly one natural number n such that n - 2 =4". Proving uniqueness The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, ''a'' and ''b'') must be equal to each other (i.e. a = b). For example, to show that the equation x + 2 = 5 has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds: : 3 + 2 = 5. To ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigidity (mathematics)
In mathematics, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c'' ∈ ''C'' is uniquely determined by less information about ''c'' than one would expect. The above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians. __FORCETOC__ Examples Some examples include: #Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values. #Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem. #By the fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existence Theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all , , ... there exist(s) ..."). In the formal terms of symbolic logic, an existence theorem is a theorem with a prenex normal form involving the existential quantifier, even though in practice, such theorems are usually stated in standard mathematical language. For example, the statement that the sine function is continuous everywhere, or any theorem written in big O notation, can be considered as theorems which are existential by nature—since the quantification can be found in the definitions of the concepts used. A controversy that goes back to the early twentieth century concerns the issue of purely theoretic existence theorems, that is, theorems which depend on non-constructive foundational material such as the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essentially Unique
In mathematics, the term essentially unique is used to describe a weaker form of uniqueness, where an object satisfying a property is "unique" only in the sense that all objects satisfying the property are equivalent to each other. The notion of essential uniqueness presupposes some form of "sameness", which is often formalized using an equivalence relation. A related notion is a universal property, where an object is not only essentially unique, but unique ''up to a unique isomorphism'' (meaning that it has trivial automorphism group). In general there can be more than one isomorphism between examples of an essentially unique object. Examples Set theory At the most basic level, there is an essentially unique set of any given cardinality, whether one labels the elements \ or \. In this case, the non-uniqueness of the isomorphism (e.g., match 1 to a or 1 to ''c'') is reflected in the symmetric group. On the other hand, there is an essentially unique ''ordered'' set of any given f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Case
In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by for rank 3 and by for rank at least 4. The three classes are groups of GF( .... The uniqueness case covers groups ''G'' of characteristic 2 type with ''e''(''G'') ≥ 3 that have an almost strongly ''p''-embedded maximal 2-local subgroup for all primes ''p'' whose 2-local ''p''-rank is sufficiently large (usually at least 3). proved that there are no finite simple groups in the uniqueness case. References * * *{{Citation , last1=Stroth , first1=Gernot , editor1-last=Arasu , editor1-first=K. T. , editor2-last=Dillon , editor2-first=J. F. , editor3-last=Harada , editor3-first=Koichiro , editor4-last=Sehgal , editor4-fir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Theorem For Poisson's Equation
The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics, this means that there is a unique electric field derived from a Electric potential, potential function satisfying Poisson's equation under the boundary conditions. __TOC__ Proof The general expression for Poisson's equation in electrostatics is :\mathbf^2 \varphi = -\frac, where \varphi is the electric potential and \rho_f is the charge density, charge distribution over some region V with boundary surface S . The uniqueness of the solution can be proven for a large class of boundary conditions as follows. Suppose that we claim to have two solutions of Poisson's equation. Let us call these two solutions \varphi_1 and \varphi_2. Then :\mathbf^2 \varphi_1 = - \frac, and :\mathbf^2 \varphi_2 = - \frac. It follows that \varphi=\varphi_2-\varphi_1 is a solution of La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thompson Uniqueness Theorem
In mathematical finite group theory, Thompson's original uniqueness theorem states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ... 3. gave a shorter proof of the uniqueness theorem. References * * * Theorems about finite groups Uniqueness theorems {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holmgren's Uniqueness Theorem
In the theory of partial differential equations, Holmgren's uniqueness theorem, or simply Holmgren's theorem, named after the Swedish mathematician Erik Albert Holmgren (1873–1943), is a uniqueness result for linear partial differential equations with real analytic coefficients. Simple form of Holmgren's theorem We will use the multi-index notation: Let \alpha=\\in \N_0^n,, with \N_0 standing for the nonnegative integers; denote , \alpha, =\alpha_1+\cdots+\alpha_n and : \partial_x^\alpha = \left(\frac\right)^ \cdots \left(\frac\right)^. Holmgren's theorem in its simpler form could be stated as follows: :Assume that ''P'' = ∑, ''α'', ≤''m'' ''A''''α''(x)∂ is an elliptic partial differential operator with real-analytic coefficients. If ''Pu'' is real-analytic in a connected open neighborhood ''Ω'' ⊂ R''n'', then ''u'' is also real-analytic. This statement, with "analytic" replaced by "smooth", is Hermann Weyl's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandrov's Uniqueness Theorem
The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s. Statement of the theorem The surface of any convex polyhedron in Euclidean space forms a metric space, in which the distance between two points is measured by the length of the shortest path from one point to the other along the surface. Within a single shortest path, distances between pairs of points equal the distances between corresponding points of a line segment of the same length; a path with this property is known as a geodesic. This property of polyhedral surfaces, that every pair ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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