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In this paper, we studied a semidiscrete coupled equation, which is integrable in the sense of admitting Lax representations. Proposed first by Vakhnenko in 2006, local conservation laws and one-fold Darboux transformation were presented with different forms, respectively, in O. O. Vakhnenko, J. Phys. Soc. Jpn. 84, 014003 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015); O. O. Vakhnenko, J. Math. Phys. 56, 033505 (2015). On the basis of these results, we principally construct N -fold Darboux transformation by means of researching gauge transformation of its Lax pair, and work out its explicit multisolutions. Given a set of seed solutions and appropriate parameters, we can calculate two-soliton solutions and plot their figures when N = 2. [ABSTRACT FROM AUTHOR]

In a recent paper, Sun posed six conjectures on the relations between T (a 1 , a 2 , ... , a k ; n) and N (a 1 , a 2 , ... , a k ; n) , where T (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 (x 1 + 1) 2 + a 2 x 2 (x 2 + 1) 2 + ⋯ + a k x k (x k + 1) 2 , where a 1 , a 2 , ... , a k are positive integers, n , x 1 , x 2 , ... , x k are arbitrary nonnegative integers, and N (a 1 , a 2 , ... , a k ; n) denotes the number of representations of n as a 1 x 1 2 + a 2 x 2 2 + ⋯ + a k x k 2 , where this time x 1 , x 2 , ... , x k are integers. In this paper, we prove Sun's six conjectures by using Ramanujan's theta function identities. [ABSTRACT FROM AUTHOR]

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a γ -Hölder continuous function 𝜃 with γ ∈ (2 3 , 1). Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to 𝜃 is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374]. [ABSTRACT FROM AUTHOR]

The homogeneous perfect sets introduced by Wen and Wu [Hausdorff dimension of homogeneous perfect sets, Acta. Math. Hungar.107 (2005) 35–44] is an important class of Moran sets. In this paper, we obtain the Assouad dimension and Assouad spectrum formulas for homogeneous perfect set under suitable condition. In the proof an Assouad spectrum formula for a large class of fractal sets is established. [ABSTRACT FROM AUTHOR]

In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The π -orbifold group of a link, Math. Z.200 (1989) 187–208], we prove that classical knots with special rank 2 are 2 -bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered. [ABSTRACT FROM AUTHOR]

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]

Let G be a simple and undirected graph with vertex set V. A set D ⊆ V is called a dominating set of G , if every vertex in V ∖ D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set of G is called the domination number of G , denoted by γ (G). A dominating set D of G is called isolate dominating, if the induced subgraph G [ D ] of G contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of G is called the isolate domination number of G , denoted by γ 0 (G). In this paper, we show that for every proper interval graph G , γ 0 (G) = γ (G). Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci.22(2) (2016) 232–241]. [ABSTRACT FROM AUTHOR]

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations [M. Reeder and J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc.27(2) (2014) 437–477, MR 3164986]. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Epipelagic L -packets and rectifying characters, Invent. Math.202(1) (2015) 1–89, MR 3402796]. The interesting feature of our computation is that we find simplifications within L -packets of the two novel invariants introduced in the above-mentioned paper of Kaletha, the toral invariant and the admissible L-embedding. [ABSTRACT FROM AUTHOR]

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p , namely the Wild and the Tame p -Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325–339], which leads to a self-contained proof for the Tame 2 -Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry. [ABSTRACT FROM AUTHOR]

Let τ be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China10 (2015) 91–110]. First they are used to obtain Chase's Theorem for τ -coherent rings. In particular, we obtain the w -version of Chase's Theorem, where w is the classical star operation in ideal theory. In the second half, we apply they to characterize τ -flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)]. [ABSTRACT FROM AUTHOR]

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys.67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal.47 (2015) 4672–4684], where the initial condition 𝜃 0 ∈ W 3 + μ , 1 (μ > 0) plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like H s . As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium. [ABSTRACT FROM AUTHOR]

Let M be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that M is η -parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in ℂ P 2 or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in ℂ H 2 . Moreover, η -parallel real hypersurfaces in ℂ P 2 and ℂ H 2 under some other conditions are classified and these results extend Suh's in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo's in [On characterizations of real hypersurfaces in a complex space form with η -parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the critical threshold phenomenon for one-dimensional damped, pressureless Euler-Poisson equations with electric force induced by a constant background, originally studied in [S. Engelberg and H. Liu and E. Tadmor, Indiana Univ. Math. J. 50 (2001) 109-157]. A simple transformation is used to linearize the characteristic system of equations, which allows us to study the geometrical structure of critical threshold curves for three damping cases: overdamped, underdamped and borderline damped through phase plane analysis. We also derive the explicit form of these critical curves. These sharp results state that if the initial data is within the threshold region, the solution will remain smooth for all time, otherwise it will have a finite time breakdown. Finally, we apply these general results to identify critical thresholds for a non-local system subjected to initial data on the whole line. [ABSTRACT FROM AUTHOR]

The goal of this paper is three-fold. First, we prove that the Cauchy problem for a generalized KP-I equation u t + | D x | α ∂ x u + ∂ x − 1 ∂ y 2 u + 1 2 ∂ x (u 2) = 0 , α ≥ 4 is locally well-posed in the anisotropic Sobolev spaces H s 1 , s 2 ( R 2) with s 1 > − α − 1 4 and s 2 ≥ 0. Second, we prove that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − (α − 1) (3 α − 4) 4 (5 α + 3) if 4 ≤ α ≤ 5. Finally, we show that the Cauchy problem is globally well-posed in H s 1 , 0 ( R 2) with s 1 > − α (3 α − 4) 4 (5 α + 4) if α > 5. Our result improves the result of Saut and Tzvetkov [The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl.79 (2000) 307–338] and Li and Xiao [Well-posedness of the fifth order Kadomtsev–Petviashvili-I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl.90 (2008) 338–352]. [ABSTRACT FROM AUTHOR]

This paper reports that synergetics are used to analyze the crazing evolution. On this basis, chaotic effect is explored. The chaos equation is established and verified. The theoretical derivation are consistent with the experimental results. We design a special specimen with a special loading mode, the transient monitoring function of acoustic emission (AE) technology is used to track and detect the crazing inside the PMMA in real time, and the experiments show that synergetics can explain the crazing properties of polymer. Importantly, the mathematical explanation is also given. The AE analysis, synergetics, and craze photo reached a conclusion that the crazing has chaotic behavior. After analyzing the AE events and crazing at different stress levels, the accuracy of synergetic approach for crazing is verified. By studying the course of AE events and crazing, the self-organization effect is proposed. The research results will provide data support for the application of PMMA in ship, aircraft, and precision instruments. [ABSTRACT FROM AUTHOR]

In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential inclusion ∇ u ∈ K subject to suitable affine boundary conditions for u with K : = SO (2) 1 δ 0 1 ∪ SO (2) 1 − δ 0 1 such that the associated deformation gradients ∇ u enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where K qc ≠ K c , and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the nonlinear matrix space geometry, it is possible to deal with the geometrically nonlinear two-well problem within the framework outlined in [A. Rüland, C. Zillinger and B. Zwicknagl, Higher Sobolev regularity of convex integration solutions in elasticity: The Dirichlet problem with affine data in int (K lc ) , SIAM J. Math. Anal.50 (2018) 3791–3841]. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures. [ABSTRACT FROM AUTHOR]

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra H , which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of H -modules is equivalent to the category H H 𝒬 𝒴 𝒟 of quasi-Yetter–Drinfeld modules. Next, we prove that H H 𝒬 𝒴 𝒟 is equivalent to the category of generalized Hopf bimodules. Finally, we show that H H 𝒬 𝒴 𝒟 is also equivalent to the category of quasi-coactions over some Majid's braided group if H is quasi-triangular. [ABSTRACT FROM AUTHOR]

In this paper, we give a recursive formula for the interesting PBW basis E A of composition subalgebras ℭ ▵ (n) of Ringel–Hall algebras ℌ ▵ (n) of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of U v + ( 𝔰 𝔩 ̂ n) from the monomial bases m (A) following [Multiplication formulas and canonical basis for quantum affine, 𝔤 𝔩 n , Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of U v + ( 𝔰 𝔩 ̂ 2) associated with modules of Loewy length ≤ 3. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras. [ABSTRACT FROM AUTHOR]

We consider an Oldroyd-B model which is derived in Ref. 4 [J. W. Barrett, Y. Lu and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci.15 (2017) 1265–1323] via micro–macro-analysis of the compressible Navier–Stokes–Fokker–Planck system. The global well posedness of strong solutions as well as the associated time-decay estimates in Sobolev spaces for the Cauchy problem are established near an equilibrium state. The terms related to η , in the equation for the extra stress tensor and in the momentum equation, lead to new technical difficulties, such as deducing L t 2 L x 2 -norm dissipative estimates for the polymer number density and its spatial derivatives. One of the main objectives of this paper is to develop a way to capture these dissipative estimates via a low–medium–high-frequency decomposition. [ABSTRACT FROM AUTHOR]

Let R ⊂ A be factorial domains containing ℚ. In this paper, we give a criterion, in terms of locally nilpotent derivations, for A to be R -isomorphic to R [ v , w ] / (c w − h (v)) , where c ∈ R is nonzero and h (v) ∈ R [ v ]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc.145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent R -derivations of the polynomial ring in two variables over R. [ABSTRACT FROM AUTHOR]

A Roman dominating function of a graph G is a labeling f : V (G) → { 0 , 1 , 2 } such that every vertex with label 0 has a neighbor with label 2. The Roman domination number, γ R (G) of G , is the minimum of ∑ v ∈ V (G) f (v) over such functions. Let G be an n -vertex graph. Chambers et al. [E. W. Chambers, B. Kinnersley, N. Prince and D. B. West, External Problems for Roman domination Siam J. Discrete Math. 23 (2009) 1575–1586.] proved that if G is a connected graph of order n ≥ 3 , then γ R (G) + γ R (G ¯) ≤ n + 3 , with equality if and only if G or G ¯ is C 5 or (n / 2) K 2 . In this paper, we construct a specific family of graphs ξ , and prove that if G ∉ ξ and G ¯ ∉ ξ , then γ R (G) + γ R (Ḡ) ≤ n + 1 , and this bound is sharp. [ABSTRACT FROM AUTHOR]

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings. [ABSTRACT FROM AUTHOR]

This paper is a continuation of [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.]. We construct by-hand Legendrian surfaces for which specific properties of their gradient flow trees hold. These properties enable us to complete the proof in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part II, to appear in Internat. J. Math.] that the Cellular DGA defined in [D. Rutherford and M. Sullivan, Cellular computation of Legendrian contact homology for surfaces, Part I, preprint (2016), arXiv:1608.02984] is stable tame isomorphic to the Legendrian contact homology DGA defined in [T. Ekholm, J. Etnyre and M. Sullivan, The contact homology of Legendrian submanifolds in ℝ 2 n + 1 , J. Differential Geom.71(2) (2005) 177–305]. [ABSTRACT FROM AUTHOR]

This paper is a continuation of [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)]. For Legendrian surfaces in 1 -jet spaces, we prove that the Cellular DGA defined in [Cellular computation of Legendrian contact homology for surfaces, preprint (2016)] is stable tame isomorphic to the Legendrian contact homology DGA, modulo the explicit construction of a specific Legendrian surface. In [Cellular computation of Legendrian contact homology for surfaces, to appear in Internat. J. Math.], we construct this surface, thereby completing Theorem 5.1 and the proof of the isomorphism. [ABSTRACT FROM AUTHOR]

Let ψ : ℕ → ℝ ≥ 0 be a non-negative function such that ψ (q) → 0 as q → ∞. The well-known Jarník–Besicovtich theorem concerns the Hausdorff dimension of the set of ψ - approximable numbers. In this paper, we give an alternative but short proof of the Jarník–Besicovitch theorem for approximating functions with no monotonicity. The main tool is the appropriate usage of the mass transference principle of Beresnevich–Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164(3) (2006) 971–992]. [ABSTRACT FROM AUTHOR]

In this paper, we establish some general Kastler–Kalau–Walze type theorems for any dimensional manifolds with boundary which generalize the results in [J. Wang and Y. Wang, The Kastler–Kalau–Walze type theorem for six-dimensional manifolds with boundary, J. Math. Phys.56 (2015), Article ID: 052501, 14 pp.]. [ABSTRACT FROM AUTHOR]

In this paper, I generalize Holzer's theorem for semi-diagonal equation. [ABSTRACT FROM AUTHOR]

In this paper, by using the Anick resolution and Gröbner-Shirshov basis for quantized enveloping algebra of type B2, we compute the minimal projective resolution of the trivial module of U q + (B 2) , and as an application we compute the global dimension of U q + (B 2). [ABSTRACT FROM AUTHOR]

Let R be a prime ring of characteristic different from 2, Q be its maximal right ring of quotients, and C be its extended centroid. Suppose that f (x 1 , ... , x n) is a non-central multilinear polynomial over C, 0   ≠   p   ∈   R , and F, G are two b-generalized derivations of R. In this paper we describe all possible forms of F and G in the case p G (F (f (r)) f (r)) = 0 for all r = (r 1 , ... , r n) in Rn. [ABSTRACT FROM AUTHOR]

In this paper, we study the (1 + 1)-dimensional Dirac equation in the presence of electric field and scalar linear potentials on (anti)-de Sitter background. Using the position representation, the energy spectrum and the corresponding wave functions are exactly obtained. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the research results of DAD, Daily Artificial Dispatcher, published in 2010 [S. Z. Stefanov, New Mathematics and Natural Computation6 (2010) 275–283], and give it new interpretation bring out the best of its meaning, as well as some more quantitative novel formula. In metaphorical sense, DAD is the analogue to DNA and hence the Smart Grid is analogue to the Creature by invoking the postmodern theory [J.-F. Lyotard, Moralités Postmodernes (Éditions Galilée, 1993) (in French)]. Specifically, the DAD’s binary expressions are generated by an innocent dialogue between DAD and Smart Grid in the form of world-strings. Namely, the living space-time of the DAD’s binary symbols is generated via a discussion between DAD and Smart Grid. Metaphorically speaking, DAD’s world is digitally described as a dramatic game and the Smart Grid as a creating cartoon loaded with an integral holographic complexity. Overall, the so called creatures capable of innocent discussions under near-zero temperature are generated by the epidemic growth of the Smart Grid cartoon. It is further concluded that the number of the Smart Grid creatures is inversely proportional to the half-time of the DAD’s binary “DNA” life cycle. Each Smart Grid can be coded in one of eight colors pool, as an aging in one of the four ways and as being in one of three possible development phases, dynamically. Finally, we take the liberty of calling the DAD’s world, a string prescribed as topology and landscape, to be “Aria”. [ABSTRACT FROM AUTHOR]

Let gcd (j , k) be the greatest common divisor of the integers j and k. In this paper, we give several interesting asymptotic formulas for weighted averages of the gcd -sum function f (gcd (j , k)) and the function ∑ d | k , d s | j (f ∗ μ) (d) for any positive integers j and k , namely ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f (gcd (j , k)) and ∑ k ≤ x 1 k s (r + 1) ∑ j = 1 k s j r ∑ d | k d s | j (f ∗ μ) (d) with any fixed integer s > 1 and any arithmetical function f. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of gcd -sum functions f (gcd (j , k)). [ABSTRACT FROM AUTHOR]

For positive integers m , n , k with m ≥ 2 and ⌈ n / 2 ⌉ ≤ k ≤ n , let E { m } (m , n , k) be the sum of multiple zeta values of depth k and weight m n with arguments m or 2 m , i.e. E { m } (m , n , k) = ∑ | α | = n α j = 1 or 2 ζ (m α 1 , m α 2 , ... , m α k). In this paper, we are going to evaluate E { m } (m , n , k). As an application, we produce the stuffle relations from k identical Riemann zeta values ζ (2 m) as well as k identical Riemann zeta values ζ (2 m) and ζ (m). [ABSTRACT FROM AUTHOR]

The classical Menon's identity [P. K. Menon, On the sum ∑ (a − 1 , n) [ (a , n) = 1 ] , J. Indian Math. Soc. (N.S.) 29 (1965) 155–163] states that ∑ a ∈ ℤ n ∗ gcd (a − 1 , n) = φ (n) σ 0 (n) , where for a positive integer n , ℤ n ∗ is the group of units of the ring ℤ n = ℤ / n ℤ , gcd (,) represents the greatest common divisor, φ (n) is the Euler's totient function and σ k (n) = ∑ d | n d k is the divisor function. In this paper, we generalize Menon's identity with Dirichlet characters in the following way: ∑ a ∈ ℤ n ∗ b 1 , ... , b k ∈ ℤ n gcd (a − 1 , b 1 , ... , b k , n) χ (a) = φ (n) σ k n d , where k is a non-negative integer and χ is a Dirichlet character modulo n whose conductor is d. Our result can be viewed as an extension of Zhao and Cao's result [Another generalization of Menon's identity, Int. J. Number Theory13(9) (2017) 2373–2379] to k > 0. It can also be viewed as an extension of Sury's result [Some number-theoretic identities from group actions, Rend. Circ. Mat. Palermo58 (2009) 99–108] to Dirichlet characters. [ABSTRACT FROM AUTHOR]

The Minkowski question mark function ? : [ 0 , 1 ] → [ 0 , 1 ] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus ? (x) is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [ N − 6 U , N + 6 U ] can be represented as n = p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 4 + p 6 4 , p i 2 − N 6 ≤ U , i = 1 , 2 , p i 3 − N 6 ≤ U , i = 3 , 4 , p i 4 − N 6 ≤ U , i = 5 , 6 , where U = N 1 − δ + 𝜀 with δ = 1 / 4 4. [ABSTRACT FROM AUTHOR]

An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to 1 is an eigenform for suitable weights on the cells. [ABSTRACT FROM AUTHOR]

In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer n, define fn:=∑k=0nnk3,Dn:=∑k=0nnk22kk2(n−k)n−k,bn:=∑k=0nnk2n+kk,An:=∑k=0nnk2n+kk2. For n=0,1,2,…, we show that 6−n|fi+j|0≤i,j≤n and 12−n|Di+j|0≤i,j≤n are positive odd integers, and 10−n|bi+j|0≤i,j≤n and 24−n|Ai+j|0≤i,j≤n are always integers. [ABSTRACT FROM AUTHOR]

Let D be a positive integer which is not a square. Further, let (u1,v1) be the least positive integer solution of the Pell equation u2−Dv2=1, and let h(4D) denote the class number of binary quadratic primitive forms of discriminant 4D. If D satisfies 2∤D and v1h(4D)≡0 (modD), then D is called an exceptional number. In this paper, under the assumption that there have no exceptional numbers, we prove that the equation xy+yx=z2 has no positive integer solutions (x,y,z) satisfy gcd(x,y)=1 and 2∤xy. [ABSTRACT FROM AUTHOR]

The modified quantum algebra associated to a quantum algebra was introduced by Lusztig. has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of for type . [ABSTRACT FROM AUTHOR]

A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained. [ABSTRACT FROM AUTHOR]

Passive synthesis of fractional driving-point immittance function is one of the most important research topics in fractional circuits. The synthesis method for a class of fractional driving-point immittance function is proposed in this paper. Firstly, the immittance function of passive fractional circuit is transformed to multivariable positive-real function by proper variable substitution. Secondly, to synthesize the function, except one variable, other high-order variables are transformed to the combination of products of different one-order variables by introducing new variables. Thirdly, synthesize the new function by traditional method. At last, substitute elements inversely twice and the final circuit for fractional driving-point immittance function can be achieved. The method is verified by giving synthesis demonstration. [ABSTRACT FROM AUTHOR]

Let be a tree of given order with diameter and the number of subtrees of . In this paper, we show that: (1) and the equality holds if and only if ; (2) If and , then and the equality holds if and only if , where (mod 3) with , and (mod 3) with ; (3) If , and (mod ) for with , then if , if , and the equalities hold if and only if . [ABSTRACT FROM AUTHOR]

In this paper, the concept of monophonic convexity is extended to weighted graphs. Set theoretic operations of monophonic convex sets and some of their properties are discussed. The notions of weighted monophonic block, weighted monophonic boundary and interior points are also studied and their characterizations are obtained. [ABSTRACT FROM AUTHOR]

A Carmichael number is a composite number such that divides for all integers coprime to . Korselt discovered that is a Carmichael number if and only if is square-free and for each prime divisor of . Let , a -number is defined to be a composite number , such that and for each prime . The set of all such that is a -number is called the Korselt set of and we denote this set by . In this paper, we investigate some properties of when is a power of a prime. [ABSTRACT FROM AUTHOR]

An -string tangle is a pair such that is a disjoint union of properly embedded arcs in a topological -ball . And an -string tangle is said to be trivial (or rational), if it is homeomorphic to as a pair, where is a 2-disk, is the unit interval and each is a point in the interior of . A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an -string stick tangle its stick-order is defined to be a nonincreasing sequence of natural numbers such that, under an ordering of the arcs of the tangle, each denotes the number of sticks constituting the th arc of the tangle. And a stick-order is said to be trivial, if every stick tangle of the order is trivial. In this paper, restricting the -ball to be the standard 3-ball, we give the complete list of trivial stick-orders. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]

Ichihara and Matsudo introduced the notions of -colorable links and the minimal coloring number for -colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable -colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable -colorable link is exactly 4. [ABSTRACT FROM AUTHOR]

In this paper, the -Gordian complex of knots is defined. The authors show that for any knot and any positive integer , there is a finite set of knots of size containing , such that the -Gordian distance between any two knots in this set is 1 for all . [ABSTRACT FROM AUTHOR]