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We quantise Whitney’s construction to prove the existence of a triangulation for any$$C^2$$C2manifold, so that we get an algorithm with explicit bounds. We also give a new elementary proof, which is completely geometric.

This paper presents an elementary proof of stochastic stability of a discrete-time reversible Markov chain starting from a Foster-Lyapunov drift condition. Besides its relative simplicity, there are two salient features of the proof: (i) it relies entirely on functional-analytic non-probabilistic arguments; and (ii) it makes explicit the connection between a Foster-Lyapunov function and Poincare inequality. The proof is used to derive an explicit bound for the spectral gap. An extension to the non-reversible case is also presented.

In (L’Enseignement Math 61(2):151–159, 2015) Nica presented an elementary proof of a result which says that the relative elementary linear group with respect to square of an ideal of a ring is a subset of the true relative elementary linear group. The original result was proved by Tits (C R Acad Sci Paris Ser A 283:693–695, 1976) in the much general context of Chevalley groups. In this paper we prove analogues of this result of Tits for transvection groups. We also obtain an elementary proof of a special case of Tits’s result, namely the case of elementary symplectic group, using commutator identities for generators of this group.

Around 1991, J. M. Borwein and P. B. Borwein introduced three cubic theta functions a(q), b(q) and c(q) and discovered many interesting identities associated with these functions. The cubic theta functions b(q) and c(q) have product representations and these representations were first established using the theory of modular forms. The first elementary proof of the product representation of b(q) was discovered in 1994 by the Borweins and F. G. Garvan using one of Euler’s identity. They then derived the product representation of c(q) using transformation formulas of Dedekind’s $$\eta (\tau )$$ and some elementary identities satisfied by a(q), b(q) and c(q). In this note, we present three proofs of the product representation of c(q) without the use of the transformation of Dedekind’s $$\eta $$ -function. We also discuss the connections between these proofs and the works of Baruah and Nath (Proc Am Math Soc 142:441–448, 2014) and Ye (Int J Number Theory 12(7):1791–1800, 2016). We also adopt the idea of the Borweins and Garvan to derive the product representation of Jacobi theta function $$\vartheta _4(0|\tau )$$ which leads to a proof of the Jacobi triple product identity.

In 19th century, Fermat's little theorem ``$a^p\equiv a({\rm mod}\;p)$ for $a\in\mathbb Z$, $p$ prime'' was generalized in two directions: Schonemann proved a corresponding congruence for the coefficients of monic polynomials, whereas Gauss found a congruence result with $p$ replaced by any $n\in\mathbb N$. Here, we shall give an elementary proof of the common generalization of these two results.

We give a short and elementary proof of the Perron–Frobenius theorem. It uses Gelfand’s formula for the spectral radius of a matrix, which is also given an elementary proof.

The halfspace depth is a prominent tool of nonparametric multivariate analysis. The upper level sets of the depth, termed the trimmed regions of a measure, serve as a natural generalization of the quantiles and inter-quantile regions to higher-dimensional spaces. The smallest non-empty trimmed region, coined the halfspace median of a measure, generalizes the median. We focus on the (inverse) ray basis theorem for the halfspace depth, a crucial theoretical result that characterizes the halfspace median by a covering property. First, a novel elementary proof of that statement is provided, under minimal assumptions on the underlying measure. The proof applies not only to the median, but also to other trimmed regions. Motivated by the technical development of the amended ray basis theorem, we specify connections between the trimmed regions, floating bodies, and additional equi-affine convex sets related to the depth. As a consequence, minimal conditions for the strict monotonicity of the depth are obtained. Applications to the computation of the depth and robust estimation are outlined.

In 1914, Felix Hausdorff published an elegant proof that almost all numbers are simply normal in base 2. We generalize this proof to show that almost all numbers are normal. The result is arguably the most elementary proof for this theorem so far and should be accessible to undergraduates in their first year., Comment: 10 pages; fixed three small errors and an incorrect argument

Let f, g be positive integers such that $$f>g$$ , $$\gcd (f,g)=1$$ and $$f\not \equiv g \pmod 2$$ . In 1956, L. Jeśmanowicz conjectured that, for any positive integer n, the equation $$(*)$$ $$\left( (f^2-g^2)n\right) ^x+(2fgn)^y=\left( (f^2+g^2)n\right) ^z$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$ . This is a far from solved problem in number theory. In 1965, V.A. Dem’janenko claimed that if $$n=1$$ and $$f=g+1$$ , then $$(*)$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$ . This result solves an important case of Jeśmanowicz’ conjecture. In this paper, on the one hand, using some properties on the representation of integers by binary quadratic primitive forms, we give a new and elementary proof for Dem’janenko’s result; on the other hand, using the Baker method and its p-adic form, we prove that if $$n>1$$ , $$f=g+1$$ and $$g=2^r$$ , where r is a positive integer with $$r\ge 80$$ and $$r+1$$ is an odd prime, then $$(*)$$ has only the positive integer solution $$(x,y,z)=(2,2,2)$$ . It can thus be seen that if $$n>1$$ , then there are infinitely many pairs (f, g) with $$f=g+1$$ which make Jeśmanowicz’ conjecture is true.

Let G be a finite group and F a field. We give an elementary proof that the group of normalized cohomological invariants of G over F with values in the Brauer group is isomorphic to $${{\,\mathrm{\mathrm {H}}\,}}^{2}(G,F^{\times })$$ .

We discuss embedding of 4-manifolds in Euclidean spaces of dimension $$ \le 8$$ . We give an elementary proof of the fact that every 4-manifold admits a topological embedding in $$\mathbb {R}^8$$ . As an application of our method of proof, we also deduce some embedding results for compact 4-manifolds with nonempty boundary.

Loomis and Whitney proved an inequality between volume and areas of projections of an open set in n-dimensional space related to the isoperimetric inequality. They reduced the problem to a combinatorial theorem proved by a repeated use of Holder inequality. In this paper we prove a general inequality between real numbers which easily implies the combinatorial theorem of Loomis and Whitney.

We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 \Delta + V(x) - E$ in dimension $n \neq 2$, where $h, \, E> 0$. The potential is real valued and $V$ and $\partial _r V$ exhibit long-range decay at infinity and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C> 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

For a field K containing 1 2 , we exhibit two matrices in the full n × n matrix algebra M n ( K ) which generate M n ( K ) as a Lie K-algebra with the commutator Lie product. We also study Lie centralizers of a not necessarily commutative unitary algebra and obtain results which we hope will eventually be a step in the direction of, firstly, proving that, for any field K, a Lie-nilpotent K-subspace (or a Lie K-subalgebra) of a finite-dimensional associative algebra over K of index k (say), generates a Lie-nilpotent associative subalgebra of much higher nilpotency index, and secondly, in consideration of the sharp upper bound for the maximum (K-)dimension of a Lie-nilpotent K-subalgebra of M n ( K ) of index k obtained in [13] , finding an upper bound for the maximum dimension of a Lie-nilpotent (of index k) Lie K-subalgebra of M n ( K ) . Finally, the constructive elementary proof of the Skolem-Noether theorem for the matrix algebra M n ( K ) in [14] , in conjunction with the well-known characterization of Lie automorphisms of M n ( K ) (if the characteristic of K is different from 2 and 3) in terms of, amongst others, automorphisms and anti-automorhisms of M n ( K ) , leads us to a unifying approach to constructively describe automorphisms and anti-automorphisms of M n ( K ) .

We give an elementary proof of a Landesman-Lazer type result for systems by means of a shooting argument and explore its connection with the fundamental theorem of algebra.

We characterize Birkhoff-James orthogonality of continuous vector-valued functions on a compact topological space. As an application of our investigation, Birkhoff-James orthogonality of real bilinear forms are studied. This allows us to present an elementary proof of the well-known Bhatia-Semrl Theorem in the real case.

Based on the Cayley-Hamilton theorem and fixed-point method, we provide an elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor in a three-dimensional (3D) inner-product space, which avoids introducing the generating function and Taylor series expansion. The proof is also extended to any finite-dimensional inner-product space.

The ring Int(Z) of integer-valued polynomials has the two-generator property, which means that every finitely generated ideal may be generated by two elements. As the known proofs of this fact are ...

In this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$ F p in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$ F p , emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$ F p -representations.

In this paper we propose an elementary proof of the Harnack inequality for linear parabolic equations in non-divergence form.

An elementary proof of Kepler's first law, i.e. that bounded planetary orbits are elliptical, is derived without the use of calculus. The proof is similar in spirit to previous derivations, in that conservation laws are used to obtain an expression for the planetary orbit, which is then compared against an equation for the ellipse. However, we derive the equation that we match against, using trigonometry, from two well-known properties of the ellipse. Calculus is avoided altogether., Comment: 7 pages, 2 figures

An elementary proof is given for the fact that every locally compact subsemigroup of a compact topological group is a closed subgroup. A sample consequence is that every commutative cancellative pseudocompact locally compact Hausdorff topological semigroup with open shifts is a compact topological group.

Quite recently, Xia and Zhao established the 10-dissections for Hirschhorn’s two infinite q-series products by using two MAPLE packages and the theory of modular forms. Utilizing the Jacobi triple product identity, we not only establish the 10-dissections for two infinite q-series products, introduced by Baruah and Kaur, but give an elementary proof of the 10-dissections due to Xia and Zhao. Moreover, we obtain the 5-dissections for four quotients of infinite q-series products related to the Rogers–Ramanujan functions. Using these dissections, the coefficients in these series expansions have periodic sign patterns with a few exceptions.

This paper provides an elementary proof of inequalities previously obtained by the operator method and having applications in additive combinatorics. The method of proof allows us to take a new look at a certain special case of Sidorenko’s conjecture.

The author presents a new approach which is used to solve an important Diophantine problem. An elementary argument is used to furnish another fully transparent proof of Fermat’s Last Theorem. This was first stated by Pierre de Fermat in the seventeenth century. It is widely regarded that no elementary proof of this theorem exists. The author provides evidence to dispel this belief.

A proof of the statement per(A ∘ B) ≤ per(A)per(B) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.

Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression for the number of roots $N_{k}(f)$. Our proof, when combined with the recent root-counting algorithm of (Dwivedi, Mittal, Saxena, CCC, 2019), yields the first deterministic poly($|f|, \log p$) time algorithm to compute $Z_{f,p}(s)$. Previously, an algorithm was known only in the case when $f$ completely splits over $\mathbb{Q}_p$; it required the rational roots to use the concept of generating function of a tree (Zuniga-Galindo, J.Int.Seq., 2003).

In this column we state a class of open problems in Ramsey Theory. The general theme is to take Ramsey-type statements that are false and weaken them by allowing the homogenous set to use more than one color. This concept is not new, and the theorems we state and/or prove are not new; however, the open questions that request easier proofs of the known theorems (or weaker versions) may be new. We use the phrase an elementary proof. This is not meant to be a technical or rigorous term. What we really mean is a proof that can be taught in an undergraduate combinatorics course. A good example of what we mean is the proof of Theorem 9.3.

An elementary proof of the no-retraction theorem in the plane is given by means of the complex square root.