Theoretical study of the stability and properties of magic numbers (m = 5, n = 2) and (m = 6, n = 3) of bimetallic bismuth‐copper nanoclusters; Bim Cun

Inspired by the experimental discovery of magic numbers we present a first study using density functional theory for the structure and properties of neutral and cationic Bi₆Cu₃ and Bi₅Cu₂ clusters. Our results confirm predictions based on Wade's rules. The closed electron shells, characteristic of cationic clusters help impose enhanced stability, while also complying with Wade's rules. Charge distribution analysis, as well as electrostatic potential maps show that in almost all cases, Bi atoms donate charges to Cu atoms. According to the analysis of condensed Fukui indices, Cu atoms inside both clusters are not reactive. Contrastingly, Bi atoms are reactive and may be targeted by different types of attack. This study of the electronic properties may thus help to determine experimental strategies with the capacity to enhance the synthesis of catalysts.     

Hua’s theorem with nine almost equal prime variables

We sharpen Hua’s result by proving that each sufficiently large odd integer N can be written as

Betti numbers of finitely presented groups and very rapidly growing functions

Define the length of a finite presentation of a group G as the sum of lengths of all relators plus the number of generators. How large can the kth Betti number b_k(G)= rank H_k(G) be providing that G has length ≤N and b_k(G) is finite? We prove that for every k≥3 the maximum b_k(N) of the kth Betti numbers of all such groups is an extremely rapidly growing function of N. It grows faster that all functions previously encountered in mathematics (outside of logic) including non-computable functions (at least those that are known to us). More formally, b_k grows as the third busy beaver function that measures the maximal productivity of Turing machines with ≤N states that use the oracle for the halting problem of Turing machines using the oracle for the halting problem of usual Turing machines.We also describe the fastest possible growth of a sequence of finite Betti numbers of a finitely presented group. In particular, it cannot grow as fast as the third busy beaver function but can grow faster than the second busy beaver function that measures the maximal productivity of Turing machines using an oracle for the halting problem for usual Turing machines. We describe a natural problem about Betti numbers of finitely presented groups such that its answer is expressed by a function that grows as the fifth busy beaver function.Also, we outline a construction of a finitely presented group all of whose homology groups are either or trivial such that its Betti numbers form a random binary sequence.     

Monotonicity Properties of Determinants of Special Functions

We prove the absolute monotonicity or complete monotonicity of some determinant functions whose entries involve

Complexity results in graph reconstruction

We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:

(1)

, , , and .

(2)

For all k?2, and .

(3)

For all k?2, .

(4)

.

(5)

For all k?2, .

For many of these results, even the c=1 case was not previously known.Similar to the definition of reconstruction numbers vrn_∃(G) [F. Harary, M. Plantholt, The graph reconstruction number, J. Graph Theory 9 (1985) 451-454] and ern_∃(G) (see [J. Lauri, R. Scapellato Topics in Graph Automorphism and Reconstruction, London Mathematical Society, Cambridge University Press, Cambridge, 2003, p. 120]), we introduce two new graph parameters, vrn_∀(G) and ern_∀(G), and give an example of a family {G_n}_n?4 of graphs on n vertices for which vrn_∃(G_n)<vrn_∀(G_n). For every k?2 and n?1, we show that there exists a collection of k graphs on (2^k-1+1)n+k vertices with 2ⁿ 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.     

Rearranged series by Haar system

For the orthonormal Haar system {X _n} the paper proves that for each 0 < ? < 1 there exist a measurable set E ? [0, 1] with measure | E | > 1 ? ? and a series of the form Σ _n=1 ^∞ a _n X _n with a _i ↘ 0, such that for every function f ∈ L ¹(0, 1) one can find a function \(\tilde f\) ∈ L ¹(0, 1) coinciding with f on E, and a series of the form

$\sum\limits_{i = 1}^\infty {\delta _i a_i \chi _i } where \delta _i = 0 or 1$

, that would converge to \(\tilde f\) in L ¹(0, 1).

     

A generalization of a curious congruence on harmonic sums

Zhao established a curious harmonic congruence for prime :

In this note the authors extend it to the following congruence for any prime and positive integer :

Other improvements on congruences of harmonic sums are also obtained.

     

New techniques for bounds on the total number of prime factors of an odd perfect number

Let denote the sum of the positive divisors of . We say that is perfect if . Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form , where are distinct primes and . Define the total number of prime factors of as . Sayers showed that . This was later extended by Iannucci and Sorli to show that . This was extended by the author to show that . Using an idea of Carl Pomerance this paper extends these results. The current new bound is .

     

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than must be divisible by a prime greater than .

     

On the smallest value of the maximal modulus of an algebraic integer

The house of an algebraic integer of degree is the largest modulus of its conjugates. For , we compute the smallest house of degree , say m. As a consequence we improve Matveev's theorem on the lower bound of m We show that, in this range, the conjecture of Schinzel-Zassenhaus is satisfied. The minimal polynomial of any algebraic integer whose house is equal to m is a factor of a bi-, tri- or quadrinomial. The computations use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in They give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer whose house is small.