Functional inequalities related to Mahler’s conjecture

Extending a result of Meyer and Reisner (Monatsh Math 125:219–227, 1998), we prove that if ${g: \mathbb{R}\to \mathbb{R}_+}$ is a function which is concave on its support, then for every m > 0 and every ${z\in\mathbb{R}}$ such that g(z) > 0, one has $$ \int\limits_{\mathbb{R}} g(x)^mdx\int\limits_{\mathbb{R}} (g^{*z}(y))^m dy\ge \frac{(m+2)^{m+2}}{(m+1)^{m+3}},$$ where for ${y\in \mathbb{R}}$ , ${g^{*z}(y)=\inf_x \frac{(1-(x-z)y)_+}{g(x)}}$ . It is shown how this inequality is related to a special case of Mahler’s conjecture (or inverse Santaló inequality) for convex bodies. The same ideas are applied to give a new (and simple) proof of the exact estimate of the functional inverse Santaló inequality in dimension 1 given in Fradelizi and Meyer (Adv Math 218:1430–1452, 2008). Namely, if ${\phi:\mathbb{R}\to\mathbb{R}\cup\{+\infty\}}$ is a convex function such that ${0 < \int e^{-\phi} < +\infty}$ then, for every ${z\in\mathbb{R}}$ such that ${\phi(z) < +\infty}$ , one has $$ \int\limits_{\mathbb{R}}e^{-\phi}\int\limits_{\mathbb{R}} e^{-\mathcal{L}^z\phi}\ge e,$$ where ${\mathcal {L}^z\phi}$ is the Legendre transform of ${\phi}$ with respect to z.     

Decomposition of cubic graphs related to Wegner’s conjecture

Thomassen formulated the following conjecture: Every 3-connected cubic graph has a red–blue vertex coloring such that the blue subgraph has maximum degree 1 (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least 1 and contains no 3-edge path. We prove the conjecture for Generalized Petersen graphs.We indicate that a coloring with the same properties might exist for any subcubic graph. We confirm this statement for all subcubic trees.     

Counterexamples to Mercat’s conjecture

In this paper, we provide counterexamples to Mercat’s conjecture on vector bundles on algebraic curves. For any \({n \geq 4}\), we provide examples of curves lying on K3 surfaces and vector bundles on those curves which invalidate Mercat’s conjecture in rank n.     

A counterexample to Valette’s conjecture

We disprove a well-known conjecture of D. Vallete (1978), which states that every d-dimensional self-affine convex body is a direct product of a polytope with a convex body of lower dimension. It is shown that there are counterexamples for dimension d = 4. Additional assumptions under which the conjecture is true are discussed.     

Some results on Vizing’s conjecture and related problems

The well-known conjecture of Vizing on the domination number of Cartesian product graphs claims that for any two graphs $G$ and $H$, $\gamma \left(G\square H\right)\ge \gamma \left(G\right)\gamma \left(H\right)$. We disprove its variations on independent domination number and Barcalkin–German number, i.e. Conjectures 9.6 and 9.2 from the recent survey Bre?ar et al. (2012) [4]. We also give some extensions of the double-projection argument of Clark and Suen (2000) [8], showing that their result can be improved in the case of bounded-degree graphs. Similarly, for rainbow domination number we show for every $k\ge 1$ that ${\gamma }_{rk}\left(G\square H\right)\ge \frac{k}{k+1}\gamma \left(G\right)\gamma \left(H\right)$, which is closely related to Question 9.9 from the same survey. We also prove that the minimum possible counterexample to Vizing’s conjecture cannot have two neighboring vertices of degree two.     

On Karatsuba’s problem related to Gram’s law

A number of new results related to Gram’s law in the theory of the Riemann zetafunction are proved. In particular, a lower bound is obtained for the number of ordinates of the zeros of the zeta-function that lie in a given interval and satisfy Gram’s law.     

Operator inequalities and reverse inequalities related to the Kittaneh–Manasrah inequalities

This work is concerned with exploring more refinement forms of the Young inequalities and the Kittaneh–Manasrah inequalities. We deduce the Operator version inequalities and reverse version inequalities related to the Kittaneh–Manasrah inequalities.     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

Integer matrices related to Liouville’s function

In this note, we construct some integer matrices with determinant equal to certain summation form of Liouville’s function. Hence, it offers a possible alternative way to explore the Prime Number Theorem by means of inequalities related to matrices, provided a better estimate on the relation between the determinant of a matrix and other information such as its eigenvalues is known. Besides, we also provide some comparisons on the estimate of the lower bound of the smallest singular value. Such discussion may be extended to that of Riemann hypothesis.     

Order of the smallest counterexample to Gallai’s conjecture

In 1966, Gallai conjectured that all the longest paths of a connected graph have a common vertex. Zamfirescu conjectured that the smallest counterexample to Gallai’s conjecture is a graph on 12 vertices. We prove that Gallai’s conjecture is true for every connected graph G with α′(G) ≤ 5, which implies that Zamfirescu’s conjecture is true.     

On De Jong’s conjecture

Let X be a smooth projective curve over a finite field F _q . Let ρ be a continuous representation π(X) → GL _n (F), where F = F _l ((t)) with F _l being another finite field of order prime to q. Assume that is irreducible. De Jong’s conjecture says that in this case is finite. As was shown in the original paper of de Jong, this conjecture follows from an existence of an F-valued automorphic form corresponding to ρ is the sense of Langlands. The latter follows, in turn, from a version of the Geometric Langlands conjecture. In this paper we sketch a proof of the required version of the geometric conjecture, assuming that char(F) ≠ 2, thereby proving de Jong’s conjecture in this case.     

Goldbach’s conjecture in max-algebra

The Goldbach conjecture is one of the best known open problems in number theory. It claims that every even integer greater than 2 can be written as the sum of two primes. The present paper formulates a max-algebraic claim that is equivalent to Goldbach’s conjecture. The max-algebraic analogue allows examination of the conjecture by the methods of max-algebra. A max-algebra is an algebraic structure in which classical addition \(+\) and multiplication \(\times \) are replaced by the operations maximum \(\oplus \) and addition \(\otimes \), in other words \(a\oplus b=\max \{a,b\}\) and \(a\otimes b=a+b\).