On generalized Stanley sequences

Let \({\mathbb{N}}\) denote the set of all nonnegative integers. Let \({k \ge 3}\) be an integer and \({A_{0} = \{a_{1}, \dots, a_{t}\} (a_{1} < \cdots < a_{t})}\) be a nonnegative set which does not contain an arithmetic progression of length k. We denote \({A = \{a_{1}, a_{2}, \ldots{}\}}\) defined by the following greedy algorithm: if \({l \ge t}\) and \({a_{1}, \dots{}, a_{l}}\) have already been defined, then \({a_{l+1}}\) is the smallest integer \({a > a_{l}}\) such that \({\{a_{1}, \dots, a_{l}\} \cup \{a\}}\) also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A₀. We prove some results about various generalizations of the Stanley sequence.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Orthogonal projections of cubes and regular simplices into a plane

We show that for every \({k\ge 2}\) and \({n\ge k}\), there is an \({n}\)-dimensional unit cube in \({\mathbb{R}^n}\) which is mapped to a regular \({2k}\)-gon by an orthogonal projection in \({\mathbb{R}^n}\) onto a \({2}\)-dimensional subspace. Moreover, by increasing dimension \({n}\), arbitrary large regular \({2k}\)-gon can be obtained in such a way. On the other hand, for every \({m\ge 3}\) and \({n\ge m-1}\), there is an \({n}\)-dimensional regular simplex of unit edge in \({\mathbb{R}^n}\) which is mapped to a regular \({m}\)-gon by an orthogonal projection onto a plane. Moreover, contrary to the cube case, arbitrary small regular \({m}\)-gon can be obtained in such a way, by increasing dimension \({n}\).     

Abelian covers of alternating groups

Let \({C={\rm inf} (k/n)\sum_{i=1}^n x_i(x_{i+1}+\cdots+x_{i+k})^{-1}}\), where the infimum is taken over all pairs of integers \({n\geq k\geq 1}\) and all positive \({x_1,\ldots,x_n}\), \({x_{n+i}=x_i}\). We prove that \({\ln 2 \leq C < 0.9305}\). In the definition of the constant C, the operation \({{\rm inf}_{k}\, {\rm inf}_{n}\, {\rm inf}_{x}}\) can be replaced by \({{\rm lim}_{k \to \infty}\, {\rm lim}_{n \to \infty} {\rm inf}_{x}}\).     

Study of LG-Holling type III predator–prey model with disease in predator

Let \(\mathbb F_{q}\) be a finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{n})\in \mathbb F_{q}^{n}\backslash \{(0,0,\ldots )\}: Tr(x_{1}^{p^{k_{1}}+1}+x_{2}^{p^{k_{2}}+1}+\cdots +x_{n}^{p^{k_{n}}+1})=c\}\), where \(c\in \mathbb F_p\), Tr is the trace function from \(\mathbb F_{q}\) to \(\mathbb F_{p}\) and each \(m/(m,k_{i})\) ( \(1\le i\le n\) ) is odd. we define a p-ary linear code \(C_{D}=\{c(a_{1},a_{2},\ldots ,a_{n}):(a_{1},a_{2},\ldots ,a_{n})\in \mathbb F_{q}^{n}\}\), where \(c(a_{1},a_{2},\ldots ,a_{n})=(Tr(a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}))_{(x_{1},x_{2},\ldots ,x_{n})\in D}\). We present the weight distributions of the classes of linear codes which have at most three weights.     

The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$

Let \(\mathbb {F}_{p^m}\) be a finite field of cardinality \(p^m\), where p is a prime, and k, N be any positive integers. We denote \(R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\) (\(u^k=0\)) and \(\lambda =a_0+a_1u+\cdots +a_{k-1}u^{k-1}\) where \(a_0, a_1,\ldots , a_{k-1}\in F_{p^m}\) satisfying \(a_0\ne 0\) and \(a_1=1\). Let r be a positive integer satisfying \(p^{r-1}+1\le k\le p^r\). First we define a Gray map from \(R_k\) to \(F_{p^m}^{p^r}\), then prove that the Gray image of any linear \(\lambda \)-constacyclic code over \(R_k\) of length N is a distance preserving linear \(a_0^{p^r}\)-constacyclic code over \(F_{p^m}\) of length \(p^rN\). Furthermore, the generator polynomials for each linear \(\lambda \)-constacyclic code over \(R_k\) of length N and its Gray image are given respectively. Finally, some optimal constacyclic codes over \(F_{3}\) and \(F_{5}\) are constructed.     

Mapping prefer-opposite to prefer-one de Bruijn sequences

We present a mapping of the binary prefer-opposite de Bruijn sequence of order n onto the binary prefer-one de Bruijn sequence of order \(n-1\). The mapping is based on the differentiation operator \(D(\langle {b_1,\ldots ,b_l}\rangle ) = \langle b_2-b_1, b_3-b_2,\ldots , b_{l}-b_{l-1} \rangle \) where bit subtraction is modulo two. We show that if we take the prefer-opposite sequence \(\langle {b_1,b_2,\ldots ,b_{2^n}}\rangle \), apply D to get the sequence \(\langle {\hat{b}_1, \ldots , \hat{b}_{2^n-1}}\rangle \) and drop all the bits \(\hat{b}_i\) such that \(\langle {\hat{b}_i,\ldots ,\hat{b}_{i+n-1}}\rangle \) is a substring of \(\langle {\hat{b}_1,\ldots ,\hat{b}_{i+n-2}}\rangle \), we get the prefer-one de Bruijn sequence of order \(n-1\).     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

Complete weight enumerators of a class of linear codes

Let \(\mathbb {F}_{q}\) be the finite field with \(q=p^{m}\) elements, where p is an odd prime and m is a positive integer. For a positive integer t, let \(D\subset \mathbb {F}^{t}_{q}\) and let \({\mathrm {Tr}}_{m}\) be the trace function from \(\mathbb {F}_{q}\) onto \(\mathbb {F}_{p}\). In this paper, let \(D=\{(x_{1},x_{2},\ldots ,x_{t}) \in \mathbb {F}_{q}^{t}\setminus \{(0,0,\ldots ,0)\} : {\mathrm {Tr}}_{m}(x_{1}+x_{2}+\cdots +x_{t})=0\},\) we define a p-ary linear code \(\mathcal {C}_{D}\) by

$$\begin{aligned} \mathcal {C}_{D}=\{\mathbf {c}(a_{1},a_{2},\ldots ,a_{t}) : (a_{1},a_{2},\ldots ,a_{t})\in \mathbb {F}^{t}_{q}\}, \end{aligned}$$

where

$$\begin{aligned} \mathbf {c}(a_{1},a_{2},\ldots ,a_{t})=({\mathrm {Tr}}_{m}(a_{1}x^{2}_{1}+a_{2}x^{2}_{2}+\cdots +a_{t}x^{2}_{t}))_{(x_{1},x_{2},\ldots ,x_{t}) \in D}. \end{aligned}$$

We shall present the complete weight enumerators of the linear codes \(\mathcal {C}_{D}\) and give several classes of linear codes with a few weights. This paper generalizes the results of Yang and Yao (Des Codes Cryptogr, 2016).

     

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.     

Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent {p > 2}

Let \({L(n)}\) be the language of group theory with n additional new constant symbols \({c_1,\ldots,c_n}\). In \({L(n)}\) we consider the class \({{\mathbb{K}}(n)}\) of all finite groups G of exponent \({p > 2}\), where \({G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z(G)}\) and \({c_1^G,\ldots,c_n^G}\) are linearly independent. Using amalgamation we show the existence of Fraïssé limits \({D(n)}\) of \({{\mathbb{K}}(n)}\). \({D(1)}\) is Felgner’s extra special p-group. The elementary theories of the \({D(n)}\) are supersimple of SU-rank 1. They have the independence property.     

Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.     

Generalizations of an Expansion Formula for Top to Random Shuffles

In the top to random shuffle, the first \({a}\) cards are removed from a deck of \({n}\) cards \({12 \cdots n}\) and then inserted back into the deck. This action can be studied by treating the top to random shuffle as an element \({B_a}\), which we define formally in Section 2, of the algebra \({{\mathbb{Q}[S_n]}}\). For \({a = 1}\), Garsia in “On the powers of top to random shuffling” (2002) derived an expansion formula for \({{B^k_1}}\) for \({{k \leq n}}\), though his proof for the formula was non-bijective. We prove, bijectively, an expansion formula for the arbitrary finite product \({B_{a1} B_{a2} \cdots B_{ak}}\) where \({a_{1}, \cdots , a_{k}}\) are positive integers, from which an improved version of Garsia’s aforementioned formula follows. We show some applications of this formula for \({B_{a1} B_{a2} \cdots B_{ak}}\), which include enumeration and calculating probabilities. Then for an arbitrary group \({G}\) we define the group of \({G}\)-permutations \({{S^G_n} := {G \wr S_n}}\) and further generalize the aforementioned expansion formula to the algebra \({{\mathbb{Q} [ S^G_n ]}}\) for the case of finite \({G}\), and we show how other similar expansion formulae in \({{\mathbb{Q} [S_n]}}\) can be generalized to \({{\mathbb{Q} [S^G_n]}}\).     

Nonlinear $$\ast$$-Lie-type derivations on von Neumann algebras

Let \({\mathcal{H}}\) be a complex Hilbert space, \({\mathcal{B(H)}}\) be the algebra of all bounded linear operators on \({\mathcal{H}}\) and \({\mathcal{A} \subseteq \mathcal{B(H)}}\) be a von Neumann algebra without nonzero central abelian projections. Let \({p_n(x_1,x_2 ,\ldots ,x_n)}\) be the commutator polynomial defined by n indeterminates \({x_1, \ldots , x_n}\) and their skew Lie products. It is shown that a mapping \({\delta \colon \mathcal{A} \longrightarrow \mathcal{B(H)}}\) satisfies

$$\delta(p_n(A_1, A_2 ,\ldots , A_n))=\sum_{k=1}^np_n(A_1 ,\ldots , A_{k-1}, \delta(A_k), A_{k+1} ,\ldots , A_n)$$

for all \({A_1, A_2 ,\ldots , A_n \in \mathcal{A}}\) if and only if \({\delta}\) is an additive *-derivation. This gives a positive answer to Conjecture 4.2 of [14].

     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Applications of strongly convergent sequences to Fourier series by means of modulus functions

The aim of this work is to estimate sums involving P(n), the largest prime factor of an integer \({n \geqq 2}\) under digital constraints \({{f(P(n)) \equiv a}{\rm mod} b}\), for every \({a \in \mathbb{Z}}\) and an integer \({b \geqq 2}\) where f is a strongly q-additive function with integer values (i.e. \({f(aq^j + b) = f(a) + f(b)}\), with \({(a, b, j) \in \mathbb{N}^3}\), \({{0 \leqq b} < q^j}\)). We also estimate the cardinality of the set \({\{{n \leqq x, f(P(n) + c)} \equiv {a {\rm mod} b}, P(n) \equiv l {\rm mod} k\}}\), where \({c \in \mathbb{Z}}\), \({k \geqq 2}\).     

The Gray images of $$$$ constacyclic codes over $$F_{2^m}[u]/\langle u^{k} \rangle $$F2m[u]/⟨uk⟩

Let \(R_{k}\) denote the polynomial residue ring \(F_{2^m}[u]/\langle u^{k} \rangle \), where \(2^{j-1}+1\le k\le 2^{j}\) for some positive integer \(j\). Motivated by the work in [1], we introduce a new Gray map from \(R_{k}\) to \(F_{2^m}^{2^{j}}\). It is proved that the Gray image of a linear \((1+u)\) constacyclic code of an arbitrary length \(N\) over \(R_{k}\) is a distance invariant linear cyclic code of length \(2^{j}N\) over \(F_{2^m}\). Moreover, the generator polynomial of the Gray image of such a constacyclic code is determined, and some optimal linear cyclic codes over \(F_{2}\) and \(F_{4}\) are constructed under this Gray map.     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

A characterization of double coverings of curves

The gonality sequence \({(d_{r})_{r}}\) of a curve X of genus g which doubly covers a curve of genus h satisfies \({d_{r} = 2(r + h)}\) for all \({r = h, h + 1, \ldots, g - 3h}\) provided that \({g \gg h}\). In this paper we explore if this striking feature of \({(d_{r})_{r}}\) actually characterizes such a covering.     

Minimal tiled orders of finite global dimension

In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers \({B_n}\) are given by the recurrence \({B_n = 6 B_{n-1} - B_{n-2}}\) with initial conditions \({B_0 = 0, B_1 = 1}\) and its associated Lucas balancing numbers \({C_n}\) are given by the recurrence \({C_n = 6 C_{n-1} - C_{n-2}}\) with initial conditions \({C_0 = 1, C_1 = 3}\). First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (x, y) for the Diophantine equation \({2x^2 + 1 = 3^b y^m}\) for any even positive integers b and m with \({m > 2}\), given in (Int J Number Theory 11:1259–1274, 2015). Also we prove that the Diophantine equations \({B_n B_{n+d}\ldots B_{n+(k-1)d} = y^m}\) and \({C_n C_{n+d}\ldots C_{n+(k-1)d} = y^m}\) have no solution for any positive integers n, d, k, y, and m with \({m \geq 2, y \geq 2}\) and gcd\({(n,d) = 1}\).