On Two Conjectures Concerning the Veronese Generating submanifolds

The notion of finite type submanifolds was introduced by B.Y.Chen.In this paper the con-jectures on scalar curvature of Veronses generating submanifolds in E^6 and the minimal conjecture on Veronese space-like submanifold ∑ and Veronese pseudo-Riemannian submanifold-↑∑ in E1^6 are proved.We have ∑ is minimal in H^5.-↑∑ is minimal in S1^5,∑ and -↑∑ are of 1-type in E1^6.     

How Neighborly Can a Centrally Symmetric Polytope Be?

We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The N-heap Wythoffs game is a two-player impartial game with N piles of tokens of sizes Players take turns removing any number of tokens from a single pile, or removing (a₁,..., a_N) from all piles - a_i tokens from the i-th pile, providing that where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N₁ such that when . ii) there exist integers N₂ and _2 such that when , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoffs game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

Symmetric Versions of Laman’s Theorem

Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are “fixed” by various symmetry operations of the framework.     

On Jemanowicz’ Conjecture Concerning Pythagorean Triples

Let(a, b, c) be a primitive Pythagorean triple. Je′smanowicz conjectured in 1956 that for any positive integer n, the Diophantine equation(an)x+(bn)y=(cn)z has only the positive integer solution(x, y, z) =(2, 2, 2). Let p ≡ 3(mod 4) be a prime and s be some positive integer. In the paper, we show that the conjecture is true when(a, b, c) =(4p2s-1, 4p s, 4p2s+ 1) and certain divisibility conditions are satisfied.     

On Ditsman’s lemma

Let H be a subgroup of a group G generated by a finite G-invariant subset X = tU _i=1 ^k C _i that consists of elements of finite order, where C _i is a class of conjugate elements of G with representative a _i . We prove that

$$|H| \leqslant \prod\limits_{i = 1}^k {o(a_i )^{|C_i |} } ,$$

where o(a _i ) is the order of the element a _i ∈ C _i . Best estimates are obtained for some important special cases.

     

On Korn’s Inequality

For β ∈ R, the authors consider the evolution system in the unknown variables u and α { ttu＋ xxxxu＋ xxtα＋（β＋｜｜ xu｜｜L2^2） xxu=f, ttα- xxα- xxtα- xxtu=0} describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.     

On Jordan’s theorem

In the present paper, we give some remarks on the well-known Jordan theorem and Hamiltonians.     

On Perelman’s dilaton

By means of a Kaluza–Klein type argument we show that the Perelman’s F{mathcal{F}} -functional is the Einstein–Hilbert action in a space with extra “phantom” dimensions. In this way, we try to interpret some remarks of Perelman in the introduction and at the end of the first section in his famous paper (Perelman in The entropy formula for the Ricci flow and its geometric applications, 2002). As a consequence the Ricci flow (modified by a diffeomorphism and a time-dependent factor) is the evolution of the “real” part of the metric under a constrained gradient flow of the Einstein–Hilbert gravitational action in higher dimension.     

On Romanov’s constant

Mathematische Zeitschrift - We show that the lower density of integers representable as a sum of a prime and a power of two is at least 0.107. We also prove that the set of integers with exactly...     

On Hasse’s inequality

We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.     

On Litoff’s theorem

We give a short proof of Litoff’s theorem from the viewpoint of completely reducible modules.