Simple Proof Of Theorem Research Articles

NOTE ON THE TWO CELEBRITY NUMBERS, PERFECT NUMBERS AND TRIANGULAR NUMBERS

Mathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. Thus a positive integer is called a perfect number if it is equal to the sum of its proper positive divisors. The search for perfect numbers began in ancient times. The four perfect numbers 6, 28, 496, and 8128 seem to have been known from ancient times. In this paper, we will investigate some important properties of perfect numbers. We give easy and simple proofs of theorems using finite series. We give our own alternative proof of the well-known Euclid’s Theorem
 (Theorem I). We will also prove some important theorems which play key roles in the mathematical theory of perfect numbers.

PROOFS FOR FINDING THE MAXIMA OF TWO QUADRATIC FORMS UNDER SOME CONSTRAINTS

A quadratic form on ℝn is a function Q defined on ℝn whose value at a vector x in ℝn is an expression of the form Q(x)=xTAx, where A is n×n symmetric matrix. In this note we discuss how to prove two theorems, Theorem A and Theorem B, for finding the maxima of two quadratic forms under some constraints. Of course, the proofs of these theorems are well known. But in this note we give an improved proof of Theorem A and a simpler proof of Theorem B.

Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger operators

This paper intends to provide new, simple, and self-contained proofs of the equivalence of various different descriptions of the uniformly hyperbolic SL (2,\mathbb R) . While in the scenario of the Schrödinger cocyles, they may in turn be applied to give new and simple proofs of theorems regarding their relation with the spectral analysis of one-dimensional discrete Schrödinger operators. Concretely, this paper gives four different descriptions of uniformly hyperbolic sequences together with the detailed proofs of their equivalence. It provides concise and self-contained proof of the Johnson’s Theorem [20], both for sequence and dynamically defined potentials. In particular, we give two different proofs of the direction “uniform hyperbolicity away from the spectrum,” of which one is the standard argument in the spirit of Russell Johnson’s original proof and the other involves Combes–Thomas type of estimate [11]. Finally, the relation between the avalanche principle, discovered by Goldstein and Schlag [16], and the uniformly hyperbolic sequence is explored which yields a simple proof and a better version of the avalanche principle. Many ingredients of the proofs in this paper are new, in particular the use of asymptotic stable and unstable directions, are of independent interest and have been applied to many other related problems.

Kalai and Muller’s possibility theorem: a simplified integer programming version

We provide a respecification of an integer programming characterization of Arrovian social welfare functions introduced by Sethuraman et al. (Math Oper Res 28:309–326, 2003). By exploiting this respecification, we give a new and simpler proof of Theorem 2 in Kalai and Muller (J Econ Theory 16:457–469, 1977).

Simple nonstandard proofs of Daniell–Kolmogorov-type theorems

This brief note employs a proof technique essentially due to Lindstrom, based on the Loeb measure construction, to obtain short and simple proofs of theorems of the Daniell–Kolmogorov type.

Addendum to “Code spectrum and the reliability function: Binary symmetric channel”

A much simpler proof of Theorem 1 from [1] is presented; we use the notation and enumeration of formulas of [1]. The text below replaces the subsection General Case in [1, Section 4, p. 271].

Remarks on subharmonic envelopes

We prove that the subharmonic envelope of a lower semicontinuous function on $\Omega$ is harmonic on a certain open subset of $\Omega$, using a very classical method in potential theory. The result gives simple proofs of theorems on harmonic measures and Jensen measures obtained by Cole and Ransford.

Fuzzy sets, fuzzy relations, and fuzzy groups: Some interrelations

We define closure operator on the set of fuzzy relations on S and show that the closed hull of a fuzzy relation ( S, μ) is given by Γ( S, μ) = ( S, μ σ μ ). We define strong fuzzy relation and derive some results to investigate the relationship between closed, strong, and symmetric fuzzy relations. We give a counterexample to show that the composition of two closed fuzzy relations need not be a closed fuzzy relation. We also give a much simpler proof of Theorem 4.2 of Bhattacharya and Mukherjee and provide counterexamples to show that their Lemma 3.4 and Theorem 3.1 of Das do not always remain true.

Block spaces on the unit sphere in 𝑅ⁿ

Let B q μ , ν B_q^{\mu ,\nu } be the block space on the unit sphere introduced by S. Lu. We discuss the relation between B q μ , ν B_q^{\mu ,\nu } and the L p {L^p} -space on the unit sphere. Then we give the characterization of B q μ , ν B_q^{\mu ,\nu } and a simple proof of Theorem (12.11)(iii) of Spaces generated by blocks (Beijing Normal University Math. Ser., 1989).

Differentiation of an operator valued measure in a Banach rigging

Remark 4. Let G be an infinite locally compact group, exPcG the subspace of all nonempty compact subsets of G. Define a mapping p :exp~O~-+~(O) taking each compact subset K to the subgroup topologically generated by K. Clearly, c(exp~G}~ ciexp G)~c~O) and p [exp~G)= (G). Therefore, if p is a continuous mapping, then c(g(G)) = c(G). It can be shown that p is continuous if and only if G is zero-dimensional and any noncompact topologically compactly generated subgroup of G is open. In particular, if G is zero-dimensional and inductively compact, then p is continuous. This yields a simpler proof of Theorem 2, but with the added assumption that G is zero-dimensional.

Compactifications of countable-dimensional and strongly countable-dimensional spaces

Simple proofs of theorems on existence of compactifications of countable-dimensional and strongly countable-dimensional spaces are given.

Simple proof of a theorem on permanents

Let A = (aij) be an n × n complex matrix. The permanent of this matrix iswhere the sum is taken over all permutations p of the set {1, …, n}.In a recent paper [1] E. H. Lieb proved an interesting theorem (see below) which he applied to verify some conjectures of M. Marcus and M. Newman. The purpose of this note is to give a simple proof of Lieb's theorem.