On a Constructive Proof of Kolmogorov’s Superposition Theorem

Kolmogorov (Dokl. Akad. Nauk USSR, 14(5):953–956, 1957) showed that any multivariate continuous function can be represented as a superposition of one-dimensional functions, i.e., $$f(x_{1},\ldots,x_{n})=\sum_{q=0}^{2n}\varPhi _{q}\Biggl(\sum_{p=1}^{n}\psi_{q,p}(x_{p})\Biggr).$$ The proof of this fact, however, was not constructive, and it was not clear how to choose the outer and inner functions Φ _q and ψ _q,p , respectively. Sprecher (Neural Netw. 9(5):765–772, 1996; Neural Netw. 10(3):447–457, 1997) gave a constructive proof of Kolmogorov’s superposition theorem in the form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ by ψ _p,q :=λ _p ψ(x _p +qa) with appropriate values λ _p ,a∈?. Basic features of this function such as monotonicity and continuity were supposed to be true but were not explicitly proved and turned out to be not valid. Köppen (ICANN 2002, Lecture Notes in Computer Science, vol. 2415, pp. 474–479, 2002) suggested a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen’s ψ, and we present a correct constructive proof of Kolmogorov’s superposition theorem for continuous inner functions ψ similar to Sprecher’s approach.     

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.     

A New Proof for the Interpolating Theorem

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A New Proof of Calabi-Yau＇s Theorem

We give a new proof of Calabi-Yau＇s theorem on the volume growth of Riemannian manifolds with non-negative Ricci curvature.     

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 Dieudonn-Manin Classification Theorem

The Dieudonn-Manin classification theorem on φ-modules (φ-isocrystals) over a perfect field plays a very important role in p-adic Hodge theory. In this note, in a more general setting we give a new proof of this result, and in the course of the proof, we also give an explicit construction of the Harder-Narasimhan filtration of a φ-module.     

On a Proof of Roth''''s Theorem

We denote by M_(n,m)(F) the set of all n×m matrices over the field F and by M_n(F) the set of all n×n matrices over the field F. W. E. Roth has shown the following theorem in 1952, [1]. Theorem Let A∈M_n(F),B∈M_m(F) and C∈M_(n,m)(F), then the matrix equation AX-YB=C (1) has a solution X, Y∈M_(n,m)(F) if and only if the matrices     

Graphs With Least Eigenvalue —2: A New Proof of the 31 Forbidden Subgraphs Theorem

Generalized line graphs were introduced by Hoffman Proc. Calgary Internat. Conf. on Combinatorial Structures and their applications, Gordon and Breach, New York (1970); they were characterized in 1980 by a collection of 31 forbidden induced subgraphs, obtained independently by Cvetkovi et al., Comptes Rendus Math. Rep. Acad. Sci. Canada (1980) and S. B. Rao et al., Proc. Second Symp., Indian Statistical Institute, Calcutta, Lecture Notes in Math., (1981). Here a short new proof of this characterization theorem is given, based on an edge-colouring technique.     

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 generalization of the category O of Bernstein–Gelfand–Gelfand

In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced a category of modules which provides a natural setting for highest weight modules. In this note, we define a family of categories which generalizes the BGG category. We classify the simple modules for some of these categories. As a consequence we show that these categories are semisimple.     

Proof of the Brunn–Minkowski Theorem by Brunn Cuts

Mathematical Notes - It is shown that for an exhaustive proof of the Brunn–Minkowski theorem on three parallel sections of a convex body, which states that if the areas of the extreme...     

Inductive Proof of a Saddle Point Theorem

Many proofs have been published for the minimax theorem, and all the published inductive proofs have been indirect ones. It has been pointed out that a direct inductive proof is needed, especially for instructional purposes, since indirect proofs are more or less implicit in nature. Such a direct proof is given in [4]: Now the minimax theorem can be stated equivalently in terms of saddle point. And it is the object of the present paper to give a direct inductive proof for the saddle point version of this theorem.     

A Note on the Pader "A Short Proof of Liapounoff''''s Convexity Theorem" of J. Lindenstrauss

J.Lindenstrauss once gave a short proof of Liapounoff＇s Convexity Theorem by using induction [1]. Now we give a more direct way to prove the theorem other than using induction. Here is the Liapounoff＇i theorem: Theorem Let μ_1,μ_2,…,μ_n,be finite positive non-atomic measures on some measure space X.Then M={(μ_1(A),μ_2(A),…,μ(A))|,measurable}is a closed and convex subset of R~n.     

The Proof of a Theorem in DC Problem

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Proof of a conjecture of Zahariuta concerning a problem of Kolmogorov on the ε-entropy

We prove a conjecture of Zahariuta which itself solves a problem of Kolmogorov on the -entropy of some classes of analytic functions. For a given holomorphically convex compact subset K in a pseudoconvex domain D in Cⁿ, Zahariutas conjecture consists in approximating the relative extremal function u^*_K,D, uniformly on any compact subset of DK, by pluricomplex Green functions on D with logarithmic poles in the compact subset K.     

A Proof of the Beyer and Stein Comjecture

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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 Generalization of the Heine�CStieltjes Theorem

The Heine?CStieltjes theorem describes the polynomial solutions, (v,f) such that T(f)=vf, to specific second-order differential operators, T, with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intricate interlacing relations between the zeros of different pairs of solutions. This extends recent results of Bourget, McMillen and Vargas for the Heun equation and answers their question of how to generalize their results to higher degrees. Many of the results are new even for the classical case.