Positive Integer Research Articles

Unlocking Your Bike the Easy Way

Combination locks are widely used to secure bicycles. We consider a combination lock consisting of adjacent rotating dials with the first nonnegative integers printed on each of them. Assuming that we know the correct combination and we start from an incorrect combination, what is the minimal number of steps to arrive at the correct combination if in each step we are allowed to turn an arbitrary number of adjacent dials once in a common direction? We answer this question using elementary methods and show how this is related to a variation of (multivariate) functions.

Conjugacy classes of completely reducible cube-free solvable p′-subgroups of GL(2,q)

Let [Formula: see text] be a cube-free positive integer and let [Formula: see text] be a prime such that [Formula: see text]. In this paper, we find the number of conjugacy classes of completely reducible solvable cube-free subgroups in [Formula: see text] of order [Formula: see text], where [Formula: see text] is a power of [Formula: see text].

On A-normaloid d-tuples of operators and related questions

Let ß A (ℍ) denote the algebra of bounded linear operators on a complex Hilbert space ℍ that admit A-adjoint operators, where A is a non-zero positive semi-definite operator on ℍ. A commuting operator tuple T = (T 1 ,…, T d ) ∈ ß A (ℍ) d is called jointly A-normaloid if rA (T) = ∥T∥ A , where rA (T) and ∥T∥ A represent the joint A-spectral radius and the joint operator A-seminorm of T, respectively. This paper aims to investigate this new class of operators and provides several examples. Furthermore, a characterization of A-normaloidity is established. Additionally, the joint Euclidean A-seminorm of a d-tuple of A-bounded operators T, denoted by , is examined. Specifically, for all positive integers n, we prove that the following equivalence holds for any commuting operator tuple . Here . Finally, several related questions are explored.

Planar Graphs without Cycles of Length 3, 4, and 6 are (3, 3)-Colorable

For non-negative integers d1 and d2, if V1 and V2 are two partitions of a graph G’s vertex set VG, such that V1 and V2 induce two subgraphs of G, called GV1 with maximum degree at most d1 and GV2 with maximum degree at most d2, respectively, then the graph G is said to be improper d1,d2-colorable, as well as d1,d2-colorable. A class of planar graphs without C3,C4, and C6 is denoted by C. In 2019, Dross and Ochem proved that G is 0,6-colorable, for each graph G in C. Given that d1+d2≥6, this inspires us to investigate whether G is d1,d2-colorable, for each graph G in C. In this paper, we provide a partial solution by showing that G is (3, 3)-colorable, for each graph G in C.

Mode recombination formula and nonanalytic term in an effective potential at finite temperature on a compactified space

We develop a new formula called a mode recombination formula, and we can recast the effective potential at finite temperature in one-loop approximation for fermion and scalar fields on the D-dimensional spacetime, Sτ1×RD−(p+1)×∏i=1pSi1 into a convenient form for discussing nonanalytic terms, which cannot be written in the form of any positive integer power of the field-dependent mass squared, in the effective potential. The formula holds irrespective of whether the field is a fermion or a scalar and of boundary conditions for spatial Si1 directions and clarifies the importance of zero modes in the Matsubara and Kaluza-Klein modes for the existence of the nonanalytic terms. The effective potential is drastically simplified further to obtain the nonanalytic terms in easier and more transparent way. In addition to reproducing previous results, we find that there exists no nonanalytic term for the fermion field with arbitrary boundary condition for the spatial Si1 direction, which is also the case for the scalar field with the antiperiodic boundary condition for the spatial direction. Published by the American Physical Society 2024

A characterization of the subspace of radially symmetric functions in Sobolev spaces

In this paper, we show that any Sobolev norm of nonnegative integer order of radially symmetric functions is equivalent to a weighted Sobolev norm of their radial profile. This establishes in terms of weighted Sobolev spaces on an interval a complete characterization of radial Sobolev spaces, which was open until now. As an application, we give a description of Sobolev norms of corotational maps.

Complexity results on k-independence in some graph products

For a positive integer k, a subset S of vertices of a graph G=(V,E) is k-independent if each vertex in S has at most k - 1 neighbors in S. We consider k-independent sets in two graph products: Cartesian and complementary prism. We show that k-independence remains NP-complete even for Cartesian products and complementary prisms. Furthermore, we present results on k-independence in grid graphs, which is a Cartesian product of two paths.

Approximate equality for two sums of roots

In this paper, we consider the problem of finding how close two sums of mth roots can be to each other. For integers m≥2, k≥1 and 0≤s≤k, let em(s,k)>0 and Em(s,k)>0 be the largest exponents such that for infinitely many integers N there exist k positive integers a1,…,ak≤N for which two sums of their mth roots ∑j=1sajm and ∑j=s+1kajm are distinct but not further than N−em(s,k) from each other, or they are distinct modulo 1 but not further than N−Em(s,k) from each other modulo 1. Some upper bounds on em(s,k) and Em(s,k) can be derived by a Liouville-type argument, while lower bounds are usually difficult to obtain. We prove that em(s,k)≥min⁡(2s,k−1,2k−2s)−1/m for 1≤s<k and that Em(s,k)≥min⁡(2s,k−2,2k−2s)+2−1/m for 0≤s≤k. Very recently, Steinerberger managed to show that E2(k,k)≥ck3, where c>0 is a small absolute constant. This seems to be the first result when the bound for s=k is increasing in k. By an entirely different argument, for any integers m≥2, k≥1 and s in the range 0≤s≤k, we show that Em(s,k)≥(k−2)/m+1. In particular, for m=2 and any non-negative integer s≤k, this yields the bound E2(s,k)≥k/2, which is much better than ck3. We also prove that e2(2,4)=7/2, which settles a problem raised by O'Rourke in 1981. These problems can be also considered for non-integer m. In particular, we show that 1≤E3/2(1,1)≤4/3, and that E3/2(1,1)=1 under assumption of the abc-conjecture.

Complete Stability of Delayed Recurrent Neural Networks With New Wave-Type Activation Functions.

Activation functions have a significant effect on the dynamics of neural networks (NNs). This study proposes new nonmonotonic wave-type activation functions and examines the complete stability of delayed recurrent NNs (DRNNs) with these activation functions. Using the geometrical properties of the wave-type activation function and subsequent iteration scheme, sufficient conditions are provided to ensure that a DRNN with n neurons has exactly (2m + 3)n equilibria, where (m + 2)n equilibria are locally exponentially stable, the remainder (2m + 3)n - (m + 2)n equilibria are unstable, and a positive integer m is related to wave-type activation functions. Furthermore, the DRNN with the proposed activation function is completely stable. Compared with the previous literature, the total number of equilibria and the stable equilibria significantly increase, thereby enhancing the memory storage capacity of DRNN. Finally, several examples are presented to demonstrate our proposed results.

Spectrality of homogeneous Moran measures on the plane

LetM=ρ1−1a0ρ2−1 be a real upper triangular expanding matrix and D=00,d10,d2d3 be a three-element real digit set with d1d3≠0 , and let {nk}k=1∞ be a sequence of positive integers with upper bound. The infinite convolutions μM,D,{nk}=δM−n1D∗δM−(n1+n2)D∗⋯∗δM−(n1+⋯+nk)D∗⋯ converges weakly to a Borel probability measure (homogeneous Moran measure). In this paper, we study the existence of exponential orthonormal basis for L2(μM,D,{nk}). A necessary and sufficient condition for μM,D,{nk} to be a spectral measure is established.

Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems

A new generalized definition of Mersenne numbers is proposed of the form an−a−1n, called global generalized Mersenne numbers and noted GMa,n with base a and exponent n positive integers. The properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, GMa,n−1 is even and divisible by n, a and a−1 for any prime n&gt;2, and by aa−1+1 for any prime n&gt;5. The remaining factor is a function of triangular numbers of a−1, specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that GMa,n≡1or7mod12 depending on the congruence of amod4; second, that GMa,n−1 are divisible by 10 if n≡1mod4 and, if n≡3mod4, GMa,n≡1or7or9mod10, depending on the congruence of amod5; third, that all factors ci of GMa,n are of the form 2nfi+1 such that ci is either prime or the product of primes of the form 2nj+1, with fi,j natural integers; fourth, that for prime n&gt;2, all GMa,n are periodically congruent to ±1or±3mod8 depending on the congruence of amod8; and fifth, that the factors of a composite GMa,n are of the form 2nfi+1 with fi≡umod4 with u=0, 1, 2 or 3 depending on the congruences of nmod4 and of amod8. The potential use of generalized Mersenne primes in cryptography is shortly addressed.

Further results on the hop domination number of a graph

A hop dominating set S in a connected graph G is called a minimal hop dominating set if no proper subset of S is a hop dominating set of G. The upper hop domination number γ+h (G) of G is the maximum cardinality of a minimal hop dominating set of G. Some general properties satisfied by this concept are studied. It is shown that for every two positive integers a and b where 2 ≤ a ≤ b, there exists a connected graph G such that γh(G) = a and γ+ h (G) = b. It is proved that minimal hop dominating set is NP-complete. It is proved that γh(G) and γ(G) are in general incomparable. It is shown that for every pair of positive integers a and b with a ≥ 2 and b ≥ 1, there exists a connected graph G such that γh(G) = a and γ(G) = b. We present an algorithm to compute minimal hop dominating set of G. Finally, we formulate an Integer linear programming problem to compute the hop domination number of G.

Katětov order between Hindman, Ramsey and summable ideals

AbstractA family $$\mathcal {I}$$ I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal $$\mathcal {I}$$ I on X is below an ideal $$\mathcal {J}$$ J on Y in the Katětov order if there is a function $$f{: }Y\rightarrow X$$ f : Y → X such that $$f^{-1}[A]\in \mathcal {J}$$ f - 1 [ A ] ∈ J for every $$A\in \mathcal {I}$$ A ∈ I . We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katětov order, where The Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), The Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), The summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent.

Each friend of 10 has at least 10 nonidentical prime factors

For each positive integer n, if the sum of the factors of n is divided by n, then the result is called the abundancy index of n. If the abundancy index of some positive integer m equals the abundancy index of n but m is not equal to n, then m and n are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10.It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10.This work applies methods from Nielsen’s 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors.This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria.

Descent of splendid Rickard equivalences in GLn(q)

Let n be a positive integer and q a prime power. We prove that a refined version of Broué's abelian defect group conjecture holds for unipotent ℓ-blocks of GLn(q), where ℓ∤q. We also give a sufficient condition on general ℓ-blocks of GLn(q) to satisfy the refined abelian defect group conjecture.

Skew linear groups with rank conditions

Let D be a division ring, n a positive integer, and GLn(D) the general skew linear group of degree n over D. In this paper, we study the structure of a subgroup G of GLn(D) satisfying some rank conditions. Among results, we prove that if G is locally nearly radical of finite co-central rank, then G has a normal solvable subgroup N of finite co-central rank such that G/N is finite. Also, we show that G is locally generalized radical of finite rank if and only if G contains a normal solvable subgroup S of finite rank such that G/S is finite. These facts generalize some our previous results.

Mixed-Input Residual Network for Air Temperature Forecasting by Low-Power Embedded Devices

Accuracy and model compactness are essential requirements for weather forecasting models designed for operation on low-power embedded devices. This study developed Mixed-Input Residual Network (MIRNet), a compact temperature-forecasting deep neural network model. MIRNet integrates stacked bidirectional long short-term memory layers using concatenated 1-dimensional and 2-dimensional convolutional layers to improve model accuracy. MIRNet was trained and tested on two datasets: one, IfeData, comprising historical weather data from Ile-Ife, Nigeria and the other a standard weather forecasting dataset called the Jena dataset. Training was carried out using 100 epochs of data partitioned in the standard 80:20 ratio, with an adaptable learning rate strategy. The model was tested for Nth-hour-ahead prediction for 1N24; where N are natural numbers, and performance quantified using metrics such as mean absolute percentage error (MAPE) and mean square error (MSE). The model was also implemented on a Raspberry Pi 4 device with a 1.8 GHz 64-bit quad-core ARM Cortex-A72 processor. The model achieved a MSE of 1.00 x 10-3 on the IfeData dataset, and 1.23x10-4 on the Jena dataset for 1-hour ahead forecasting. This is currently the best verifiable result achieved on the Jena dataset by any prediction model globally. For Nth hour ahead forecasting, MIRNet achieved an MSE generally below 2.0x10-3 for all values of N on the standard Jena dataset. The MSE of MIRNet for N-sequential 1-hour ahead and single Nth hour predictions using the Jena dataset reveal quadratic and linear relationships with N respectively. The model compares favourably with existing models for multi-hour predictions. The developed model is compact and has good forecasting properties.

Properties of arithmetics progressions in increasing sequence of T0-topologies on the set of positive integers

In this paper we continue researches deal with increasing sequence {Tm} of T0-topologies on the set of positive integers focusing on the properties of arithmetic progressions in topologies Tm. We characterize the closures of arithmetic progressions in all topologies Tm on N. Additionally, we present the characterization of the closures of arithmetic progressions in the common division topology T on N. Moreover, for each m∈N we characterize regular open arithmetic progressions in (N,Tm) and we examine which of these spaces are semiregular.

Solving the membership problem for certain subgroups of [formula omitted]

For positive integers u and v, let Lu=[10u1] and Rv=[1v01]. Let Gu,v be the group generated by Lu and Rv. In a previous paper, the authors determined a characterization of matrices M=[abcd] in Gu,v when u,v≥3 in terms of the short continued fraction representation of b/d. We extend this result to the case where uv≥5. Additionally, we compute [Gu,v:Gu,v] where Gu,v={[1+uvn1vn2un31+uvn4]∈SL2(Z):(n1,n2,n3,n4)∈Z4} for u,v≥1, extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for Gu,v with u,v≥1.

The [formula omitted]-dimensional bootstrap percolation models with axial neighbourhoods

Fix positive integers d,r and a1≤a2≤⋯≤ad. For large L, each site of {1,…,L}d⊂Zd can be at state 0 or 1 (infected), and its neighbourhood consists of the ak nearest neighbours in the ±ek-directions for each k∈{1,2,…,d}. The state will evolve in discrete time as follows: At time 0, vertices are independently 1 with some probability p. We infect any vertex v∈{1,…,L}d at state 0 already having r infected neighbours, and infected sites remain infected forever.In this paper we study the critical length for percolation, defined by Lc(Nra1,…,ad,p)=min{L∈N:Pp({1,…,L}dis eventually infected)≥1/2}. We determine the (d−1)-times iterated logarithm of Lc(Nra1,…,ad,p) up to a constant factor, for all d-tuples (a1,…,ad) and all r∈{a2+⋯+ad+1,…,a1+a2+⋯+ad}.We conjecture that we can reduce the problem of determining this threshold for all d≥3 and all r∈{ad+1,…,a1+a2+⋯+ad}, to that of determining the threshold for all d≥3 and r∈{ad+1,…,ad−1+ad} only.