Distinct subset sums and an inequality for convex functions

In this note we prove an inequality for convex functions which implies a conjecture of P. Erdos about a finite integer set with distinct subset sums.     

A problem on partial sums in abelian groups

In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group that naturally arises investigating simple Heffter systems. Then we show its connection with related open problems and we present some results about the validity of these conjectures.     

Integer sets with distinct subset sums

We give a simple, elementary new proof of a generalization of the following conjecture of Paul Erdos: the sum of the elements of a finite integer set with distinct subset sums is less than 2.

     

Constancy of functions restricted to a subset

Consider a function from a finite set to the real line. Under certain conditions the function becomes a constant function when restricted to a subset. This has important combinatorial significance. In fact an important conjecture of Wang [1] is proved and extended considerably using the result.     

New results on the conjecture of Rhodes and on the topological conjecture

The Conjecture of Rhodes, originally called the “type II conjecture” by Rhodes, gives an algorithm to compute the kernel of a finite semigroup. This conjecture has numerous important consequences and is one of the most attractive problems on finite semigroups. It was known that the conjecture of Rhodes is a consequence of another conjecture on the finite group topology for the free monoid. In this paper, we show that the topological conjecture and the conjecture of Rhodes are both equivalent to a third conjecture and we prove this third conjecture in a number of significant particular cases.     

Feasibility of the Mixed Postman Problem with Restrictions on the Edges

The mixed postman problem consists of finding a minimum cost tour of a connected mixed graph traversing all its vertices, edges, and arcs at least once. We consider the variant of the mixed postman problem where all edges must be traversed exactly once. The feasibility version of this problem is NP-complete. We introduce an infinite class of necessary conditions for feasibility, which we conjecture are also sufficient. We prove that no finite subset of these conditions is sufficient.     

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics, and Fourier analysis.      

Tate''s Conjecture, Algebraic Cycles and Rational K-Theory in Characteristic p

The purpose of this article is to discuss conjectures on motives, algebraic cycles and K-theory of smooth projective varieties over finite fields. We give a characterization of Tate's conjecture in terms of motives and their Frobenius endomorphism. This is used to prove that if Tate's conjecture holds and rational and numerical equivalence over finite fields agree, then higher rational K-groups of smooth projective varieties over finite fields vanish (Parshin's conjecture). Parshin's conjecture in turn implies a conjecture of Beilinson and Kahn giving bounds on rational K-groups of fields in finite characteristic. We derive further consequences from this result.     

Cutsets and anti-chains in linear lattices

Consider the poset, ordered by inclusion, of subspaces of a four-dimensional vector space over a field with 2 elements. We prove that, for this poset, any cutset (i.e., a collection of elements that intersects every maximal chain) contains a maximal anti-chain of the poset. In analogy with the same result by Duffus, Sands, and Winkler [D. Duffus, B. Sands, P. Winkler, Maximal chains and anti-chains in Boolean lattices, SIAM J. Discrete Math. 3 (2) (1990) 197-205] for the subset lattice, we conjecture that the above statement holds in any dimension and for any finite base field, and we prove some special cases to support the conjecture.     

A Conjecture on the Hall Topology for the Free Group

The Hall topology for the free group is the coarsest topologysuch that every group morphism from the free group onto a finitediscrete group is continuous. It was shoen by M.Hall Jr thatevery finitely generated subgroup of the free group is closedfor this topology. We conjecture that if H₁, H₂,...,H_n are finitelygenerated subgroups of the free group, then the product H₁ H₂...H_n is closed. We discuss some consequences of this conjecture.First, it would give a nice and simple algorithm to computethe closure of a given rational subset of the free group. Next,it implies a similar conjecture for the free monoid, which inturn is equivalent to a deep conjecture on finite semigroupsfor the solution of which J. Rhodes has offered $100. We hopethat our new conjecture will shed some light on Rhodes' conjecture.     

A numerical approach to proving projectivity

Summary We present a nonconstructive method which uses intersection numbers and linear space theory for proving the existence of projective embeddings of suitable algebraic schemes, and we apply it to establish Chevalley's conjecture that a complete nonsingular variety such that any finite number of points is contained in an open affine subset is projective. In memory of Guido Castelnuovo in the recurrence of the first centenary of his birth.     

Finiteness conjecture and subdivision

The finiteness conjecture by J.C. Lagarias and Y. Wang states that the joint spectral radius of a finite set of square matrices is attained on some finite product of such matrices. This conjecture is known to be false in general. Nevertheless, we show that this conjecture is true for a big class of finite sets of square matrices used for the smoothness analysis of scalar univariate subdivision schemes with finite masks.     

Cyclic polytopes and d-order curves

A cyclic d-polytope is a convex polytope combinatorially equivalent to the convex hull of a finite subset of a d-order curve in R ^d. We give an affirmative answer to a conjecture of M. A. Perles [4] by proving that every even-dimensional cyclic polytope occurs in this way: its set of vertices can always be extended to a d-order curve.     

The Tate conjecture for K3 surfaces over finite fields

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.     

A proof of the strong no loop conjecture

The strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. The main result of this paper establishes this well known conjecture for finite dimensional algebras over an algebraically closed field.     

Aggregation via empirical risk minimization

Given a finite set F of estimators, the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in F. It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the convex hull of a well chosen, empirically determined subset of F is an optimal aggregation method.     

Normal sequences over finite abelian groups

In this note, we obtain the structure of short normal sequences over a finite abelian p-group or a finite abelian group of rank two, thus answering positively a conjecture of Gao and Zhuang for various groups. The results obtained here improve all known results on this conjecture.     

A singular function with a non-zero finite derivative

This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its generalizations and variants—the derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative 1 on a subset of the Cantor set.     

New families of balanced symmetric functions and a generalization of Cusick,Li and Stǎnicǎ’s conjecture

In this paper we provide new families of balanced symmetric functions over any finite field. We also generalize a conjecture of Cusick, Li, and Stǎnicǎ about the non-balancedness of elementary symmetric Boolean functions to any finite field and prove part of our conjecture.     

On 3rd and 4th moments of finite upper half plane graphs

Terras [A. Terras, Fourier Analysis on Finite Groups and Applications, Cambridge Univ. Press, 1999] gave a conjecture on the distribution of the eigenvalues of finite upper half plane graphs. This is known as a finite analogue of Sato–Tate conjecture. There are several modified versions of them. In this paper, we show that this conjecture is not correct in its original form (i.e., Conjecture 1.1). This is shown for the calculations of the 3rd and 4th moments of the distribution of the eigenvalues. We remark that a weaker version of the conjecture (i.e., Conjecture 1.2) may still hold.