New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane

In the projective planes PG(2, q), more than 1230 new small complete arcs are obtained for ${q \leq 13627}$ and ${q \in G}$ where G is a set of 38 values in the range 13687,..., 45893; also, ${2^{18} \in G}$ . This implies new upper bounds on the smallest size t ₂(2, q) of a complete arc in PG(2, q). From the new bounds it follows that $$t_{2}(2, q) < 4.5\sqrt{q} \, {\rm for} \, q \leq 2647$$ and q = 2659,2663,2683,2693,2753,2801. Also, $$t_{2}(2, q) < 4.8\sqrt{q} \, {\rm for} \, q \leq 5419$$ and q = 5441,5443,5449,5471,5477,5479,5483,5501,5521. Moreover, $$t_{2}(2, q) < 5\sqrt{q} \, {\rm for} \, q \leq 9497$$ and q = 9539,9587,9613,9623,9649,9689,9923,9973. Finally, $$t_{2}(2, q) <5 .15\sqrt{q} \, {\rm for} \, q \leq 13627$$ and q = 13687,13697,13711,14009. Using the new arcs it is shown that $$t_{2}(2, q) < \sqrt{q}\ln^{0.73}q {\rm for} 109 \leq q \leq 13627\, {\rm and}\, q \in G.$$ Also, as q grows, the positive difference ${\sqrt{q}\ln^{0.73} q-\overline{t}_{2}(2, q)}$ has a tendency to increase whereas the ratio ${\overline{t}_{2}(2, q)/(\sqrt{q}\ln^{0.73} q)}$ tends to decrease. Here ${\overline{t}_{2}(2, q)}$ is the smallest known size of a complete arc in PG(2,q). These properties allow us to conjecture that the estimate ${t_{2}(2,q) < \sqrt{q}\ln ^{0.73}q}$ holds for all ${q \geq 109.}$ The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms. Finally, new forms of the upper bound on t ₂(2,q) are proposed.     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume $4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}$ . A 1-isometry of the central quadric $\mathcal{F}: = \{ x \in V|q(x) = 1\}$ is a permutation ? of $\mathcal{F}$ such that (*) $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ holds true for a fixed element ν of $\mathbb{F}$ . For arbitraryν ∈ $\mathbb{F}$ we prove that? is induced (in a certain sense) by a semi-linear bijection $(\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})$ such thatq oσ =? oq, provided $\mathcal{F}$ contains lines and the exceptional case $(\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)$ is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.     

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in {\mathbb {R}^{3}}

We consider a class of semilinear elliptic equations of the form $$ \label{eq:abs}-\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\mathbb {R}^{3},$$ where ${a:\mathbb {R} \to \mathbb {R}}$ is a periodic, positive, even function and, in the simplest case, ${W : \mathbb {R} \to \mathbb {R}}$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W^{\prime}(q(x))=0,\ x\in\mathbb {R},\quad q(x)\to\pm1\,{\rm as}\, x\to \pm\infty,$$ we show, via variational methods the existence of infinitely many geometrically distinct solutions u of (0.1) verifying u(x, y, z) → ± 1 as x → ± ∞ uniformly with respect to ${(y, z) \in \mathbb {R}^{2}}$ and such that ${\partial_{y}u \not \equiv0, \partial_{z}u \not\equiv 0}$ in ${\mathbb {R}^{3}}$ .     

Poisson kernel and Cauchy formula of a non-symmetric transitive domain

In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain S m,p,q={Z∈Cm×m, Z1∈Cm×p,Z2 ∈Cq×m|2i1( Z-Z+)-Z1Z1′-Z2′Z20} does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when S m,p,q is not symmetric. However the map T0:Z→Z, Z1→Z1 , Z2→Z2 transforms S m,p,q into a domain S I (m, m + p + q) which can be mapped by the Cayley transformation into the classical domains R I (m, m + p + q). The pull back of the Bergman metric of R I (m, m + p + q) to S m,p,q is a Riemann metric ds 2 which is not a Khler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator associated with the metric ds 2 when it acts on the Poisson kernel of S m,p,q equals 0. Consequently, the Cauchy formula of S m,p,q can be obtained from the Poisson formula.     

Generalised Gagliardo–Nirenberg Inequalities Using Weak Lebesgue Spaces and BMO

Using elementary arguments based on the Fourier transform we prove that for ${1 \leq q < p < \infty}$ and ${s \geq 0}$ with s > n(1/2 ? 1/p), if ${f \in L^{q,\infty} (\mathbb{R}^n) \cap \dot{H}^s (\mathbb{R}^n)}$ , then ${f \in L^p(\mathbb{R}^n)}$ and there exists a constant c _p,q,s such that $$\| f \|_{L^{p}} \leq c_{p,q,s} \| f \|^\theta _{L^{q,\infty}} \| f \|^{1-\theta}_{\dot{H}^s},$$ where 1/p = θ/q + (1?θ)(1/2?s/n). In particular, in ${\mathbb{R}^2}$ we obtain the generalised Ladyzhenskaya inequality ${\| f \| _{L^4} \leq c \| f \|^{1/2}_{L^{2,\infty}} \| f \|^{1/2}_{\dot{H}^1}}$ .We also show that for s = n/2 and q > 1 the norm in ${\| f \|_{\dot{H}^{n/2}}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon–Zygmund decompositions.     

New open-book decompositions in singularity theory

In this article, we study the topology of real analytic germs ${F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)}$ given by ${F(x,y,z)=\overline{xy}(x^p+y^q)+z^r}$ with ${p,q,r \in \mathbb{N}, p,q,r \geq 2}$ and (p, q)?=?1. Such a germ gives rise to a Milnor fibration ${\frac{F}{\mid F \mid}\colon \mathbb{S}^5\setminus L_F \to \mathbb{S}^1}$ . We describe the link L _F as a Seifert manifold and we show that in many cases the open-book decomposition of ${\mathbb{S}^5}$ given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ${\mathbb{C}^3}$ .     

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σ _d|n d. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.     

On the permutability of Backlund transformations

Let(?)=B_ηu:2(q-(?))+(⊿((?)-2q))+(2q_x+(?)_x))η=0,2(r-(?)+(⊿(2(?)-r)+(r_x+2(?)_x))η=0,u=(q,r)~Tbe the Backlund transformation (BT) of the hierarchy of AKNS equations,where η is a parameterand Δ=integral from -∞ to x (qr-(?))dx′.It is shown in this paper the infinitesimal BT B_(η+ε)B_η~(-1) admits thefollowing expansionB_(η+ε)B_η~(-1)u=u+εsum from n=0 to ∞ β_n(JL~(n+1)u)η~n,β_n=1+(-1)~n2~(-n-1),where L is the recurrence operator of the hierarchy and ε is an infinitesimal parameter.Thisexpansion implies the equivalence between the permutabiliy of BTs and the involution in pairs ofconserved densities.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

Integrability of vector-valued lacunary series with applications to function spaces

Let ${\mathcal L(r) = \sum_{n=0}^\infty a_nr^{\lambda_n}}$ be a lacunary series converging for 0 <  r < 1, with coefficients in a quasinormed space. It is proved that $$\int_0^1 F(1-r,\|\mathcal L(r)\|)(1-r)^{-1}\,{\rm d}r < \infty $$ if and only if $$ \sum_{n=0}^\infty F(1/\lambda_n,\|a_n\|) < \infty, $$ where F is a “normal function” of two variables. In the case when p ≥ 1 and F(x, y) =  x y ^p , this reduces to a theorem of Gurariy and Matsaev. As an application we prove that if ${f(r\zeta) = \sum_{n=0}^\infty r^{\lambda_n}f_{\lambda_n}(\zeta)}$ is a function harmonic in the unit ball of ${\mathbb R^N,}$ then $$\int_0^1M_p^q(r,f)(1-r)^{q\alpha-1} \,{\rm d}r <\infty\quad (p,\,q,\,\alpha >0 ) $$ if and only if $$\sum_{n=0}^\infty \|f_{\lambda_n} \|^q_{L^p(\partial B_N)}(1/\lambda_n)^{q\alpha} <\infty. $$      

Existence of homoclinic solutions for second order Hamiltonian systems with general potentials

In this paper we are concerned with the existence of homoclinic solutions for the following second order non-autonomous Hamiltonian systems HS $$ \ddot{q}-L(t)q+W_{q}(t,q)=0, $$ where $W\in C^{1}(\mathbb{R}\times\mathbb{R}^{n},\mathbb{R})$ and $L\in C(\mathbb{R},\mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in\mathbb{R}$ . Assuming that the potential W satisfies some weaken global Ambrosetti-Rabinowitz conditions and L meets the coercive condition, we show that (HS) has at least one nontrivial homoclinic solution via using the Mountain Pass Theorem. Some recent results in the literature are generalized and significantly improved.     

The equality problem in the class of conjugate means

Let ${I\subset\mathbb{R}}$ be a nonempty open interval and let ${L:I^2\to I}$ be a fixed strict mean. A function ${M:I^2\to I}$ is said to be an L-conjugate mean on I if there exist ${p,q\in{]}0,1]}$ and a strictly monotone and continuous function φ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q)\varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all ${x,y\in I}$ . Here L(x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means.     

Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

Maximal, potential and singular operators in vanishing generalized Morrey spaces

We introduce vanishing generalized Morrey spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\Omega), \Omega \subseteq \mathbb{R}^n}$ with a general function ${\varphi(x, r)}$ defining the Morrey-type norm. Here ${\Pi \subseteq \Omega}$ is an arbitrary subset in Ω including the extremal cases ${\Pi = \{x_0\}, x_0 \in \Omega}$ and Π = Ω, which allows to unify vanishing local and global Morrey spaces. In the spaces ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n)}$ we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type ${V\mathcal{L}^{p,\varphi}_\Pi (\mathbb{R}^n) \rightarrow V\mathcal{L}^{q,\varphi^\frac{q}{p}}_\Pi (\mathbb{R}^n)}$ -theorem for the potential operator I ^α . The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ${\varphi(x, r)}$ . No monotonicity type condition is imposed on ${\varphi(x, r)}$ . In case ${\varphi}$ has quasi- monotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ${\varphi}$ . The proofs are based on pointwise estimates of the modulars defining the vanishing spaces     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .