Robin’s criterion on divisibility

The Ramanujan Journal - Robin’s criterion states that the Riemann hypothesis is true if and only if the inequality $$\sigma (n) < e^{\gamma } \times n \times \log \log n$$ holds for...     

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.     

A Note on Toeplitz’ Conjecture

In 1911, Toeplitz made a conjecture asserting that every Jordan curve in $\mathbb{R}^{2}$ contains four points forming the corners of a square. Here Conjecture C is presented, which states that the side length of the largest square on a closed curve that consists of edges of an n×n grid is at least $1/\sqrt{2}$ times the side length of the largest axis-aligned square contained inside the curve. Conjecture C implies Toeplitz’ conjecture and is verified computationally for n≤13.     

Schmidt’s game on fractals

We construct (α, β) and α-winning sets in the sense of Schmidt’s game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some. In particular, we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of ℝ ^N satisfying the open set condition, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then for any countable collection of non-singular affine transformations, Δ _i : ℝ ^N → ℝ ^N ,

Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

Some remarks on Kong-Liu’s conjecture on Lorentzian metrics

For a(1+3)-dimensional Lorentzian manifold(M,g),the general form of solutions of the Einstein field equations takes that of type I,II,or III.For type I,there is a known result in Gu(2007).In this paper,we try to find the necessary and sufficient conditions for the Lorentzian metric to take the form of types II and III,and we show how to construct the new coordinate system.     

Dirac’s Theorem on Simplicial Matroids

We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k = 2) and they are a rich new class of simplicial complexes. We show that Dirac’s theorem on chordal graphs has a higher-dimensional analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and k-hyperclique complexes. We prove also a higher-dimensional analogue of Stanley’s reformulation of Dirac’s theorem on chordal graphs.      

Brendle’s Inequality on Static Manifolds

We generalize Brendle’s geometric inequality considered in Brendle (Publ Math Inst Hautes Études Sci 117:247–269, 2013) to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperbolic manifolds in the spirit of Chru?ciel and Simon (J Math Phys 42(4):1779–1817, 2001).     

A Note on Schnyder’s Theorem

We give an alternate proof of Schnyder’s Theorem, that the incidence poset of a graph G has dimension at most three if and only if G is planar.     

A remark on Gagola’s Theorem

In this paper, we generalize Gagola’s Theorem [1]. Firstly we obtain several new identities. With the help of these identities, we prove a conclusion similar with Gagola’s under some more general conditions. Finally, we get a result regarding the control of p-transfer by Tate’s Theorem.     

Remarks on Serre’s modularity conjecture

In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof does not use any modularity lifting theorem in characteristic 2 (moreover, we will not consider at all characteristic 2 representations at any step of our proof). The key tool in the proof is a very general modularity lifting result of Kisin, which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved.     

A note on Gekeler’s h-function

For a subgroup L of the symmetric group \({S_{\ell}}\), we determine the minimal base size of \({GL_d(q) \wr L}\) acting on \({V_d(q)^{\ell}}\) as an imprimitive linear group. This is achieved by computing the number of orbits of GL_d(q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of V_m(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of \({GL_{d}(q) \wr L}\) satisfy a conjecture of Pyber concerning bases.     

Newton’s method on Lie groups

The purpose of this paper is to study the strong convergence of fixed points for a family of demi-continuous pseudo-contractions by hybrid projection algorithms in the framework of Hilbert spaces. Our results improve and extend the corresponding results announced by many others.     

A note on Barnette’s conjecture

A well-known conjecture of Barnette states that every 3-connected cubic bipartite planar graph has a Hamiltonian cycle, which is equivalent to the statement that every 3-connected even plane triangulation admits a 2-tree coloring, meaning that the vertices of the graph have a 2-coloring such that each color class induces a tree. In this paper we present a new approach to Barnette’s conjecture by using 2-tree coloring.A Barnette triangulation is a 3-connected even plane triangulation, and a B-graph is a smallest Barnette triangulation without a 2-tree coloring. A configuration is reducible if it cannot be a configuration of a B-graph. We prove that certain configurations are reducible. We also define extendable, non-extendable and compatible graphs; and discuss their connection with Barnette’s conjecture.     

Short note on Hilbert’s inequality

Considering five different parameters, we obtain some new Hilbert-type integral inequalities for functions f(x), g(x) in L²[0, ∞). Then, we extract from our results some special cases which have been proved before.     

Jensen’s inequality on convex spaces

We prove a Jensen’s inequality on $p$ -uniformly convex space in terms of $p$ -barycenters of probability measures with $(p-1)$ -th moment with $p\in ]1,\infty [$ under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT $(\kappa )$ -space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into $2$ -uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.