A New Proof for the Interpolating Theorem

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A Simple Proof for a Theorem of Kelisky and Rivlin

Theorem (Kelisky and Rivlin) Let f(x) be a function defined in [0,1] and B_n(f(x))=sum from k=o to n (f(k/s)(?)x~k(1-x)~(n-k)) be the nth Bernstein polynomial of f(x). Then lim B~l(f(x))=f(0)+(f(1)-f(0))x. Proof We can assume f(0)=0, Let φ_i(x) and ψ_i(x)(i=1,2,…,n) be Bernstein basis polynomials and Bezier basis polynomials respectively. Let n×n matrices     

A construction of D-optimal designs for N≡2 mod 4

D-optimal design of order 6 is used to construct D-optimal designs of order 42 and 66.     

A New Proof of the Assmus–Mattson Theorem for Non-Binary Codes

C. Bachoc gave a new proof of theAssmus–Mattson theorem for linear binary codes using harmonicweight enumerators which she defined B. We give a new proof ofthe Assmus–Mattson theorem for linear codes over any finitefield using similar methods.     

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let {M³_i}\{M^{3}_{i}\} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0 . Suppose that all unit metric balls in M³_iM^{3}_{i} have very small volume, at most v _i →0 as i→∞, and suppose that either M³_iM^{3}_{i} is closed or has possibly convex incompressible toral boundary. Then M³_iM^{3}_{i} must be a graph manifold for sufficiently large i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.     

A Simple Proof for a Property of the Vector Space Ⅴ Over the Field F

<正> In this paper we'll prove a fundameutal property of the vector space by means of the ex-tension field,i.e.the numbers of the elements of a basis of the vector space V over the field Fequal to the d imensions(V:F).     

A Simple Proof of Regularity for C~(1,α) Interface Transmission Problems

We give an alternative proof of a recent result in [1] by Caffarelli,Soria-Carro, and Stinga about the C~(1,α)regularity of weak solutions to transmission problems with C~(1,α)interfaces. Our proof does not use the mean value property or the maximum principle, and also works for more general elliptic systems with variable coefficients. This answers a question raised in [1]. Some extensions to C~(1,Dini) interfaces and to domains with multiple sub-domains are also discussed.     

Interval Approach to Phase Measurements Can Lead to Arbitrarily Complex Sets – A Theorem and Ways around It

We are often interested in phases of complex quantities; e.g., in nondestructive testing of aerospace structures, important information comes from phases of Pulse Echo and magnetic resonance. For each measurement, we have an upper bound Δ on the measurement error Δx= ${\tilde x}$ ?x, so when the measurement result is ${\tilde x}$ , we know that the actual value x is in [ ${\tilde x}$ ?Δ, ${\tilde x}$ +Δ]. Often, we have no information about probabilities of different values, so this interval is our only information about x. When the accuracy is not sufficient, we perform several repeated measurements, and conclude that x belongs to the intersection of the corresponding intervals. For real-valued measurements, the intersection of intervals is always an interval. For phase measurements, we prove that an arbitrary closed subset of a circle can be represented as an intersection of intervals. Handling such complex sets is difficult. It turns out that if we have some statistical information, then the problem often becomes tractable. As a case study, we describe an algorithm that uses both real-valued and phase measurement results to determine the shape of a fault. This is important: e.g., smooth-shaped faults gather less stress and are, thus, less dangerous than irregularly shaped ones.     

A priori estimates and blow-up behavior for solutions of −QNu = Veu in bounded domain in ℝN

Let Q_N be an N-anisotropic Laplacian operator, which contains the ordinary Laplacian operator, N-Laplacian operator and the anisotropic Laplacian operator. We firstly obtain the properties of Q_N, which contain the weak maximal principle, the comparison principle and the mean value property. Then a priori estimates and blow-up analysis for solutions of Q_Nu in bounded domain in ?^N, N ≥ 2 are established. Finally, the blow-up behavior of the only singular point is also considered.     

A New Proof of Diophantine Equation （^n2） = （^m4）

By using algebraic number theory and p-adic analysis method, we give a new and simple proof of Diophantine equation （^n2） = （^m4）     

Stability Estimates for Coefficients of Magnetic Schrödinger Equation from Full and Partial Boundary Measurements

In this paper we establish a log log-type estimate which shows that in dimension n ≥ 3 the magnetic field and the electric potential of the magnetic Schrödinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary ?Ω – a notion made more precise later. Furthermore, we prove that in the case when the measurement is taken on all of ?Ω one can establish a better estimate that is of log-type.     

Rigorous Numerics for Dissipative Partial Differential Equations II. Periodic Orbit for the Kuramoto–Sivashinsky PDE—A Computer-Assisted Proof

We present a method of self-consistent a priori bounds, which allows us to study rigorously the dynamics of dissipative PDEs. As an application we present a computer-assisted proof of the existence of a periodic orbit for the Kuramoto-Sivashinsky equation where = 0.127.     

A Markoff-type chain for the view-obstruction problem for spheres in ℝ4

In the view-obstruction problem, congruent, closed, convex bodies centred at the points in ⁿ are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size. In the case of spheres of diameter 1, this value is denoted byv(n) and is known for dimensionsn=2,3. Here we show that and obtain a Markoff type chain of isolated extreme values.