Complex periodical sequences with lower autocorrelation values are used in CDMA communication systems and cryptography. In this paper we present new nonexistence results on perfect p-ary sequences and almost p-ary sequences and related difference sets by using some knowledge on cyclotomic fields and their subfields.

Given a domain Ω ⊂ Rⁿ, let λ > 0 be an eigenvalue of the elliptic operator L :=Σ_i,jⁿ =1 ∂/∂_xj(a^ij ∂/∂_xj) on Ω for Dirichlet condition. For a function ƒ ∈ L²(Ω), it is known that the linear resonance equation Lu + λu = ƒ in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition P_λ(u|∂Ω) = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ‖u‖_2,2 ≤ C(‖ƒ‖₂ +‖g‖_2,2) under suitable regularity assumptions on ∂Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2,p-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean (Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.

The Conway potential function (CPF) for colored links is a convenient version of the multivariable Alexander-Conway polynomial. We give a skein characterization of CPF, much simpler than the one by Murakami. In particular, Conway's “smoothing of crossings” is not in the axioms. The proof uses a reduction scheme in a twisted group-algebra P_nB_n, where Bn is a braid group and P_n is a domain of multi-variable rational fractions. The proof does not use computer algebra tools. An interesting by-product is a characterization of the Alexander-Conway polynomial of knots.

Recently, Cristofaro-Gardiner and Hutchings proved that there exist at least two closed characteristics on every compact star-shaped hypersuface in R⁴. Then Ginzburg, Hein, Hryniewicz, and Macarini gave this result a second proof. In this paper, we give it a third proof by using index iteration theory, resonance identities of closed characteristics and a remarkable theorem of Ginzburg et al.

We determine the maximum order E_g of finite groups G acting on the closed surface Σ_g of genus g which extends over (S³,Σ_g) for all possible embeddings Σ_g → S³, where g > 1.

We study space-like self-shrinkers of dimension n in pseudo-Euclidean space R_m^m+n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.

We give a summary on the recent development of chaos theory in topological dynamics, focusing on Li-Yorke chaos, Devaney chaos, distributional chaos, positive topological entropy, weakly mixing sets and so on, and their relationships.

We first determine in this paper the structure of the generalized Fitting subgroup F^*(G) of the finite groups G all of whose defect groups (of blocks) are conjugate under the automorphism group Aut(G) to either a Sylow p-subgroup or a fixed p-subgroup of G. Then we prove that if a finite group L acts transitively on the set of its proper Sylow p-intersections, then either L/O_p(L) has a T.I. Sylow p-subgroup or p = 2 and the normal closure of a Sylow 2-subgroup of L/O₂(L) is 2-nilpotent with completely descripted structure. This solves a long-open problem. We also obtain some generalizations of the classic results by Isaacs and Passman on half-transitivity.

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.

Let L=-div(A▽) be a second order divergence form elliptic operator, and A be an accretive, n×n matrix with bounded measurable complex coefficients in Rⁿ. We obtain the L^p bounds for the commutator generated by the Kato square root √L and a Lipschitz function, which recovers a previous result of Calderón, by a different method. In this work, we develop a new theory for the commutators associated to elliptic operators with Lipschitz function. The theory of the commutator with Lipschitz function is distinguished from the analogous elliptic operator theory.

By means of Fourier frequency localization and Bony's paraproduct decomposition, we study the global existence and the uniqueness of the 2D Leray-α Magneta-hydrodynamics model with zero magnetic diffusivity for the general initial data. In view of the profits bring by the α model, then using the energy estimate in the frequency space and the Logarithmic Sobolev inequality, we obtain the estimate ∫₀^t||▽u||_L∞ds which is crucial to get the global existence for the general initial data.

Given a graph G with n vertices and an Abelian group A of order n, an A-distance antimagic labelling of G is a bijection from V(G) to A such that the vertices of G have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of A) of the labels assigned to its neighbours. An A-distance magic labelling of G is a bijection from V(G) to A such that the weights of all vertices of G are equal to the same element of A. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.

Let L be the derivation Lie algebra of C[t₁^±1,t₂^±1]. Given a triangle decomposition L=L⁺⊕h⊕L^-, we define a nonsingular Lie algebra homomorphism ψ: L⁺→C and the universal Whittaker L-module W_ψ of type ψ. We obtain all Whittaker vectors and submodules of W_ψ. Moreover, all simple Whittaker L-modules of type ψ are determined.

Let f(z) be a finite order meromorphic function and let c ∈ C\{0} be a constant. If f(z) has a Borel exceptional value a ∈ C, it is proved that max{τ(f(z)), τ(Δ_cf(z))}=max{τ(f(z)), τ(f(z+c))}=max{τ(Δ_cf(z)), τ(f(z+c))}=σ(f(z)). If f(z) has a Borel exceptional value b ∈ (C\{0}) ∪ {∞}, it is proved that max {τ(f(z)), τ(Δ_cf(z)/f(z))}=max{τ (Δ_cf(z)/f(z), τ(f(z+c))}=σ(f(z)) unless f(z) takes a special form. Here τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).

Chen and Zhang [Sci. China, Ser. A, 45, 1390-1397 (2002)] introduced an affine scaling trust region algorithm for linearly constrained optimization and analyzed its global convergence. In this paper, we derive a new affine scaling trust region algorithm with dwindling filter for linearly constrained optimization. Different from Chen and Zhang's work, the trial points generated by the new algorithm are accepted if they improve the objective function or improve the first order necessary optimality conditions. Under mild conditions, we discuss both the global and local convergence of the new algorithm. Preliminary numerical results are reported.

In this short note, we consider the perturbation of compact quantum metric spaces. We first show that for two compact quantum metric spaces (A, P) and (B, Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively, the quantum Gromov-Hausdorff distance between (A, P) and (B, Q) is small under certain conditions. Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.

In this paper, we study some functional inequalities (such as Poincaré inequality, logarithmic Sobolev inequality, generalized Cheeger isoperimetric inequality, transportation-information inequality and transportation-entropy inequality) for reversible nearest-neighbor Markov processes on connected finite graphs by means of (random) path method. We provide estimates of the involved constants.

In this paper, the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied. There are three approaches to solve this problem. The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths; the second approach is to find a current assignment of the current graph by the theory of current graph; the third approach is to find exponentially many embedding (or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph. According to this three approaches, we can construct exponentially many minimum genus embeddings of complete graph K_12s+8 in orientable surfaces, which show that there are at least 10/3×(200/9)^s distinct minimum genus embeddings for K_12s+8 in orientable surfaces. We have also proved that K_12s+8 has at least 10/3×(200/9)^s distinct minimum genus embeddings in non-orientable surfaces.

In this paper, we characterize the sharp boundedness of the one-sided fractional maximal function for one-weight and two-weight inequalities. Also a new two-weight testing condition for the one-sided fractional maximal function is introduced extending the work of Martín-Reyes and de la Torre. Improving some extrapolation result for the one-sided case, we get weak sharp bounded estimates for one-sided fractional maximal function and weak and strong sharp bounded estimates for one-sided fractional integral.

In this paper, the authors establish the boundedness of commutators generated by strongly singular Calderón-Zygmund operators and weighted BMO functions on weighted Herz-type Hardy spaces. Moreover, the corresponding results for commutators generated by strongly singular Calderón-Zygmund operators and weighted Lipschitz functions can also be obtained.

Let X denote a compact metric space with distance d and F:X×R→X or F_t:X→X denote a C⁰-flow. From the point of view of ergodic theory, all important dynamical behaviors take place on a full measure set. The aim of this paper is to introduce the notion of Banach upper density recurrent points and to show that the closure of the set of all Banach upper density recurrent points equals the measure center or the minimal center of attraction for a C⁰-flow. Moreover, we give an example to show that the set of quasi-weakly almost periodic points can be included properly in the set of Banach upper density recurrent points, and point out that the set of Banach upper density recurrent points can be included properly in the set of recurrent points.

By using the stable t-structure induced by an adjoint pair, we extend several results concerning recollements to upper (resp. lower) recollements. These include the fundamental results of Parshall and Scott on comparisons of recollements, Wiedemann's result on the global dimension and Happel's result on the finitistic dimension, occurring in a recollement (D^b(A'),D^b(A),D^b(A″)) of bounded derived categories of Artin algebras. We introduce and describe a triangle expansion of a triangulated category and illustrate it by examples.

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every 1-planar graph G with maximum degree Δ(G) ≥ 12 and girth at least five is totally (Δ(G)+1)-colorable.

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al.[J. Differential Geom., 99, 359-409(2015)]. In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.

Let C be a triangulated category with a proper class ε of triangles. We prove that there exists an Avramov-Martsinkovsky type exact sequence in C, which connects ε-cohomology, ε-Tate cohomology and ε-Gorenstein cohomology.

Let A be an expansive dilation on Rⁿ and φ:Rⁿ×[0, ∞)→[0, ∞) an anisotropic Musielak-Orlicz function. Let H_A^φ(Rⁿ) be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. In this article, the authors establish its molecular characterization via the atomic characterization of H_A^φ(Rⁿ). The molecules introduced in this article have the vanishing moments up to order s and the range of s in the isotropic case (namely, A:=2I_n×n) coincides with the range of well-known classical molecules and, moreover, even for the isotropic Hardy space H^p(Rⁿ) with p∈(0, 1] (in this case, A:=2I_n×n, φ(x, t):=t^p for all x∈Rⁿ and t∈[0, ∞)), this molecular characterization is also new. As an application, the authors obtain the boundedness of anisotropic Calderón-Zygmund operators from H_A^φ(Rⁿ) to L^φ(Rⁿ) or from H_A^φ(Rⁿ) to itself.

We develop a new method to perturb singular circuits in configuration Space B with trivial isotropy groups, and construct a homomorphism Φ* between its circuit cobordism groups and pseudohomology groups. This work can be viewed as a counterpart to Zinger's construction.

In this paper, we investigate the intersection numbers of nearly Kirkman triple systems. J_N[v] is the set of all integers k such that there is a pair of NKTS(v)s with a common uncovered collection of 2-subset intersecting in k triples. It has been established that J_N[v]={0, 1,..., (v(v-2))/(6)-6,(v(v-2))/(6)-4,(v(v-2))/(6)} for any integers v≡0 (mod 6) and v≥66. For v≤60, there are 8 cases left undecided.

Let γ be a hyperbolic closed orbit of a C¹ vector field X on a compact C^∞ manifold M of dimension n≥3, and let H_X(γ) be the homoclinic class of X containing γ. In this paper, we prove that C¹-generically, if H_X(γ) is expansive and isolated, then it is hyperbolic.

Let G be a finite group and let N be a nilpotent normal subgroup of G such that G/N is cyclic. It is shown that under some conditions all Coleman automorphisms of G are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at the origin.

In this paper, for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs, a novel eccentricity-based invariant, named as non-self-centrality number (NSC number for short), of a graph G is defined as follows:N(G)=Σ_vi,v_j∈V(G)|e_i-e_j|where the summation goes over all the unordered pairs of vertices in G and e_i is the eccentricity of vertex v_i in G, whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G. In this paper, we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained. Finally we propose some attractive research topics for this new invariant of graphs.

In this paper, we mainly explore fractal dimensions of fractional calculus of continuous functions defined on closed intervals. Riemann-Liouville integral of a continuous function f(x) of order v(v>0) which is written as D^-vf(x) has been proved to still be continuous and bounded. Furthermore, upper box dimension of D^-vf(x) is no more than 2 and lower box dimension of D^-vf(x) is no less than 1. If f(x) is a Lipshciz function, D^-vf(x) also is a Lipshciz function. While f(x) is differentiable on[0, 1], D^-vf(x) is differentiable on[0, 1] too. With definition of upper box dimension and further calculation, we get upper bound of upper box dimension of Riemann-Liouville fractional integral of any continuous functions including fractal functions. If a continuous function f(x) satisfying Hölder condition, upper box dimension of Riemann-Liouville fractional integral of f(x) seems no more than upper box dimension of f(x). Appeal to auxiliary functions, we have proved an important conclusion that upper box dimension of Riemann-Liouville integral of a continuous function satisfying Hölder condition of order v(v>0) is strictly less than 2-v. Riemann-Liouville fractional derivative of certain continuous functions have been discussed elementary. Fractional dimensions of Weyl-Marchaud fractional derivative of certain continuous functions have been estimated.

The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space. Motivated by the numerical results of Hairer and Voss, we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.

Positive entire solutions of the equation Δ_pu=u^-q in R^N (N≥2) where 1 < p≤N, q>0, are classified via their Morse indices. It is seen that there is a critical power q=q_c such that this equation has no positive radial entire solution that has finite Morse index when q>q_c but it admits a family of stable positive radial entire solutions when 0 < q≤q_c. Proof of the stability of positive radial entire solutions of the equation when 1 < p < 2 and 0 < q≤q_c relies on Caffarelli-Kohn-Nirenberg's inequality. Similar Liouville type result still holds for general positive entire solutions when 2 < p≤N and q>q_c. The case of 1 < p < 2 is still open. Our main results imply that the structure of positive entire solutions of the equation is similar to that of the equation with p=2 obtained previously. Some new ideas are introduced to overcome the technical difficulties arising from the p-Laplace operator.

In this paper, the authors consider the problem of existence of periodic solutions for a second order neutral functional differential system with nonlinear difference D-operator. For such a system, since the possible periodic solutions may not be differentiable, our method is based on topological degree theory of condensing field, not based on Leray Schauder topological degree theory associated to completely continuous field.