Volume 68, Issue 5

We state and prove the “First law of Cubology”, i.e. the solvability criterion, for the n × n × n Rubik's Cube.

In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r , ∑i+j+k=pri,j,k∈Pp1i⁢j⁢k≡-2⁢pr-1⁢Bp-3(modpr),$$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where Pn$ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n . In this note, we obtain the congruences for distinct odd primes p , q and positive integers α , β , ∑i+j+k=pα⁢qβi,j,k∈P2⁢p⁢q1i⁢j⁢k≡78⁢(2-q)⁢(1-1q3)⁢pα-1⁢qβ-1⁢Bp-3(modpα)$$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and ∑i+j+k=pα⁢qβi,j,k∈Pp⁢q(-1)ii⁢j⁢k≡12⁢(q-2)⁢(1-1q3)⁢pα-1⁢qβ-1⁢Bp-3(modpα).$$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

By a sphere-packing argument, we show that there are infinitely many pairs of primes that are close to each other for some metrics on the integers. In particular, for any numeration basis q , we show that there are infinitely many pairs of primes the base q expansion of which differ in at most two digits. Likewise, for any fixed integer t , there are infinitely many pairs of primes, the first t digits of which are the same. In another direction, we show that, there is a constant c depending on q such that for infinitely many integers m there are at least c log log m primes which differ from m by at most one base q digit.

Let D ∈ ℤ and let C D be the set of all monic cubic polynomials with integer coefficients having a discriminant equal to D . In this paper, we devise a general method of establishing whether, for a prime p , all polynomials in C D have the same type of factorization over the Galois field Fp$\Bbb F_p$.

In this paper, we study abstract convexity topical and sub-topical functions. We obtain some results on abstract convexity such as support set and subdifferential in view of new elementary functions. Indeed, first we show that the ICR and IR functions are dense in IPH functions then the topical function and sub-topical function on topological vector space by another elementary function is studied. These elementary functions lead to obtain similar results with easier proofs.

We introduce new classes of functions related to metric-preserving functions and b-metrics. We investigate their properties and compare them to those of metric-preserving functions.

In this paper we establish some related Ostrowski inequalities for the case of convex functions and general Lebesgue integral on measurable spaces. Midpoint and integral mean inequalities are provided, some particular results related to the famous Fejér’s inequality are also given.

For the σ -ideal N$\mathcal{N} $ of nullsets and σ -ideal M$\mathcal{M} $ of microscopic sets, it was recently obtained that there exists a Borel set E⊂ℝ2$E\subset\mathbb{R}^{2} $ with the following property: Ex∈M$E_{x}\in\mathcal{M} $ for any x∈ℝ$x\in\mathbb{R} $ and {y;Ey∉N}∉M$\{y;E^{y}\notin\mathcal{N}\}\notin\mathcal{M} $, for vertical sections Ex={y;(x,y)∈E}$E_{x}=\{y;(x,y)\in E\} $ and horizontal sections Ey={x;(x,y)∈E}$E^{y}=\{x;(x,y)\in E\} $ for E⊂ℝ2$E\subset\mathbb{R}^{2} $. Thus (N,M)$(\mathcal{N},\mathcal{M}) $ does not satisfy Fubini Property. In this paper we obtain such Borel set E , that {y;Ey∉N}∉M$\{y;E^{y}\notin\mathcal{N}\}\notin\mathcal{M} $ and all non-horizontal (in a natural sense) sections of E are in M$\mathcal{M} $. Other Fubini type properties, with conditions written for all directions are also discussed.

The main object of the present paper is to investigate a number of useful properties such as sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness and radius of convexity for a new subclass of meromorphic multivalent starlike functions, which are defined here by means of a newly defined q- linear differential operator.

In this paper, we consider a new direct approach namely the tan(φ(ξ)/2)-Expansion Method to obtain analytical solutions of Burgers and foam drainage equations. With this method, further solutions can be obtained comparing with other techniques and approaches. We use of Mathematica for obtaining these solutions.

Modeling ecological and environmental problems lead to the study of mathematical systems with missing data. In this paper we study a parameter identification question of a pollution problem using the method of sentinels by Lions (1992). Our system, which is a spatio-temporal one contains a missing initial condition. We show that the question of existence of a sentinel insensitive to the initial condition is equivalent to an approximate null-controllability problem for which we present a variational method.

We study some conditions of solvability of a semilinear system in ℝ n , where the linear part is represented by an n × n matrix with one-dimensional kernel and the nonlinear term is a sublinear, continuous vector field.

The purpose of this paper is to derive sufficient conditions for the oscillation of all solutions of a difference equation with several non-monotone deviating arguments and nonnegative coefficients. Corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the significance of the results are also given.

In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators ℳβ+$ \mathcal{M}_{\beta}^{+} $ and ℳβ-$ \mathcal{M}_{\beta}^{-} $ map W1,p⁢(ℝ)$ W^{1,p}(\mathbb{R}) $ into W1,q⁢(ℝ)$ W^{1,q}(\mathbb{R}) $ with 1 < p <∞, 0≤β<1/ p and q = p /(1- pβ ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators Mβ+$ M_{\beta}^{+} $ and Mβ-$ M_{\beta}^{-} $ from ℓ1⁢(ℤ)$ \ell^{1}(\mathbb{Z}) $ to BV⁢(ℤ)$ {\rm BV}(\mathbb{Z}) $. Here BV⁢(ℤ)$ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

In this note we give very short proofs for Banach contraction principle theorem in the b -rectangular metric spaces and b -metric spaces. Our result provides a complete solution to an open problem raised by George, Radenović, Reshma and Shukla.

Let V denote the classical Volterra operator on L 2 [0,1] and let z 1 , z 2 be arbitrary complex numbers. We investigated the operator norm of z1⁢V+z2⁢V*$ z_{1}V+z_{2}V^{*} $.

In this paper, we present several numerical radius inequalities for Hilbert space operators. More precisely, we prove if T,U∈BH$ T,U\in\mathbb{B}\left(\mathcal{H}\right) $ such that U is unitary, then ω⁢(T⁢U±U*⁢T)≤2⁢ω⁢(T2)+∥T±T*∥2.$$\displaystyle\omega(TU\pm U^{*}T)\leq 2\sqrt{\omega(T^{2})+\|T\pm T^{*}\|^{2}}. $$ Also, we have compared our results with some known outcomes.

We investigate four-dimensional CR submanifolds in six-dimensional strict nearly Kähler manifolds. We construct a moving frame that nicely corresponds to their CR structure and use it to investigate CR submanifolds that admit a special type of doubly twisted product structure. Moreover, we single out a class of CR submanifolds containing this type of doubly twisted submanifolds. Further, in a particular case of the sphere S6⁢(1)$ \mathbb{S}^{6}(1) $, we show that the two families of four-dimensional CR submanifolds, those that admit a three-dimensional geodesic distribution and those ruled by totally geodesic spheres S3$ \mathbb{S}^{3} $ coincide, and give their classification, which as a subfamily contains a family of doubly twisted CR submanifolds.

Let X be a continuum. The n -fold hyperspace C n ( X ), n < ∞, is the space of all nonempty closed subsets of X with at most n components. A topological property P$ \mathcal{P} $ is said to be a (an almost) sequential decreasing strong size property provided that if μ is a strong size map for C n ( X , {tj}j=1∞$ \{t_{j}\}_{j=1}^{\infty} $ is a sequence in the interval ( t ,1) such that lim t j = t ∈ [0,1) ( t ∈ (0,1)) and each fiber μ −1 ( t j ) has property P$ \mathcal{P} $, then so does μ −1 ( t ). In this paper we show that the following properties are sequential decreasing strong size properties: being a Kelley continuum, local connectedness, continuum chainability and, unicoherence. Also we prove that indecomposability is an almost sequential decreasing strong size property.

In this paper we give a partial response to one of the most important statistical questions, namely, what optimal statistical decisions are and how they are related to (statistical) information theory. We exemplify the necessity of understanding the structure of information divergences and their approximations, which may in particular be understood through deconvolution. Deconvolution of information divergences is illustrated in the exponential family of distributions, leading to the optimal tests in the Bahadur sense. We provide a new approximation of I -divergences using the Fourier transformation, saddle point approximation, and uniform convergence of the Euler polygons. Uniform approximation of deconvoluted parts of I -divergences is also discussed. Our approach is illustrated on a real data example.

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason’s theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner’s theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces A of a Hilbert space H${\mathcal H}$ correspond to direct product decompositions H≃A×A⊥$\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. In [10] it was shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner’s theorem: for an infinite set X , the automorphism group of the orthomodular poset Fact( X ) of direct product decompositions of X is isomorphic to the permutation group of X . The structure Fact( X ) plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact( X ) are determined in proving our version of Wigner’s theorem, namely that Fact( X ) is atomistic in a very strong way.

In the present paper, we prove that a prime ring R with center Z satisfies s 4 , the standard identity in four variables if R admits a non-identity automorphism σ such that ( u σ , u ] v σ + v σ [ u σ , u ]) n ∈ Z for all u , v in some non-central Lie ideal L of R whenever either char( R )> n or char( R )=0, where n is a fixed positive integer.