Chen’ s theorem in arithmetical progressions

LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_2 ,p \equiv l_1 (\bmod q), P_2 \equiv l_2 (\bmod q)$$ has infinitely many solutions for almost all $q \leqslant N^{\frac{1}{{37}}} $ , wherep is a prime andP ₂ is an almost prime with at most two prime factors.     

Relative widths of function classes of L 2(T) defined by a linear differential operator in L q (T)

In this paper, the smallest number M which makes the equality $$ K_n (W_2^{L_r } (T),MW_2^{L_r } (T),L_2 (T)) = d_n (W_2^{L_r } (T),L_2 (T)) $$ valid, is established and the asymptotic order of $$ K_n (W_2^{L_r } (T),W_2^{L_r } (T),L_q (T)),1 \leqslant q \leqslant \infty $$ , is obtained, where $ W_2^{L_r } $ (T) is a periodic smooth function class which is determined by a linear differential operator, K _n (·, ·, ·) and d _n (·, ·) are the relative width and the width in the sense of Kolmogorov, respectively.     

On the average number of direct factors of a finite Abelian group

Let \(T(x) = \sum\limits_{ord(G) \leqq x} {t(G),} \) , wheret(G) define the number of direct factors of a finite Abelian group.E. Krätzel ([5]) defined a remainderΔ ₁(x) in the asymptotic ofT(x) and proved $$\Delta _1 (x)<< x^{{5 \mathord{\left/ {\vphantom {5 {12}}} \right. \kern-\nulldelimiterspace} {12}}} \log ^4 x.$$ Using two different methods to estimate a special three-dimensional exponential sum we get the better results $$\Delta _1 (x)<< x^{{{282} \mathord{\left/ {\vphantom {{282} {683}}} \right. \kern-\nulldelimiterspace} {683}}} \log ^4 x$$ and $$\Delta _1 (x)<< x^{{{45} \mathord{\left/ {\vphantom {{45} {109}}} \right. \kern-\nulldelimiterspace} {109}} + \varepsilon } (\varepsilon > 0).$$      

An asymptotic property of the number of spanning trees of double fixed step loop networks

Let T(C(p,1,q))be tbe number of spanning trees of circulant graph C(p,1,q). In this note we show that T(C(p 1,1,q))/T(C(p,1,q))→2,p→ ∞.     

Regular prism tilings in {{\widetilde{\bf SL_{2}R}}} space

\({{\widetilde{\bf SL_{2}R}}}\) geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all 2 × 2 real matrices with determinant one. Our aim is to describe and visualize the regular infinite or bounded p-gonal prism tilings in \({{\widetilde{\bf SL_{2}R}}}\) . For this purpose we introduce the notion of infinite and bounded prisms, prove that there exist infinitely many regular infinite p-gonal face-to-face prism tilings \({\mathcal{T}^i_p(q)}\) and infinitely many regular bounded p-gonal non-face-to-face \({{\widetilde{\bf SL_{2}R}}}\) prism tilings \({\mathcal{T}_p(q)}\) for integer parameters \({p,q; 3 \leq p, \frac{2p}{p-2} < q}\) . Moreover, we describe the symmetry group of \({\mathcal{T}_p(q)}\) via its index 2 rotational subgroup, denoted by pq2 ₁ . Surprisingly this group already occurred in our former work (Molnár et al., J Geometry, 95:91–133, 2009) in another context. We also develop a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to \({\mathcal{T}^i_3(q)}\) and \({\mathcal{T}_3(q)}\) where 6 < q and visualize them and the corresponding tilings. E. Molnár showed, that homogeneous 3-spaces have a unified interpretation in the projective 3-sphere \({\mathcal{PS}^3}\) and 3-space \({\mathcal{P}^3({\bf V}^4,{\bf V}_4, {\bf R})}\) . In our work we will use this projective model of \({{\widetilde{\bf SL_{2}R}}}\) and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of a computer.     

A New Bound on the Number of Designs with Classical Affine Parameters

The hyperplanes in the affine geometry AG(d, q) yield an affineresolvable design with parameters $2 - (q^d ,q^{d - 1} ,\frac{{q^{d - 1} - 1}}{{q - 1}})$ . Jungnickel [3]proved an exponential lower bound on the number of non-isomorphic affine resolvable designs with these parametersfor d ≥ 3. The bound of Jungnickel was improved recently [5] by a factor of $q^{\frac{{d^2 + d - 6}}{2}} (q - 1)^{d - 2}$ for any d ≥ 4. In this paper, a construction of $2 - (q^d ,q^{d - 1} ,\frac{{q^{d - 1} - 1}}{{q - 1}})$ designs based on group divisible designs is given that yieldsat least $$\frac{{\left( {q^{d - 1} + q^{d - 2} + \cdots + 1} \right)!\left( {q - 1} \right)}}{{\left| {{\text{P}}\Gamma {\text{L(}}d,q{\text{)}}} \right|\left| {{\text{A}}\Gamma {\text{L(}}d,q{\text{)}}} \right|}}$$ non-isomorphic designs for any d ≥ 3. This new bound improves the bound of[5] by a factor of $$\frac{1}{{q^d }}\mathop \Pi \limits_{i = 1}^{(q^{d - 1} - q)/(q - 1)} (q^{d - 1} + i).$$ For any given q and d, It was previously known [2,11] that there are at least 8071non-isomorphic 2-(27,9,4) designs. We show that the number of non-isomorphic 2-(27,9,4) is atleast 245,100,000.     

On the Oscillation of nth Order Dynamic Equations on Time-Scales

We present some new criteria for the oscillation of even order dynamic equation $$\left(a(t)({x^\Delta}^{n-1}(t))^\alpha\right)^\Delta +q(t)(x^\sigma(t))^\lambda = 0$$ on an unbounded above time scale ${\mathbb{T}}$ , where α and λ are the ratios of positive odd integers, a and q is a real valued positive rd-continuous functions defined on ${\mathbb{T}}$ .     

An integral equation and a representation for a Green's function

The final step in the mathematical solution of many problems in mathematical physics and engineering is the solution of a linear, two-point boundary-value problem such as $$\begin{gathered} \ddot u - q(t)u = - g(t), 0< t< x \hfill \\ (0) = 0, \dot u(x) = 0 \hfill \\ \end{gathered} $$ Such problems frequently arise in a variational context. In terms of the Green's functionG, the solution is $$u(t) = \int_0^x {G(t, y, x)g(y) dy} $$ It is shown that the Green's function may be represented in the form $$G(t,y,x) = m(t,y) - \int_y^x {q(s)m(t, s) m(y, s)} ds, 0< t< y< x$$ wherem satisfies the Fredholm integral equation $$m(t,x) = k(t,x) - \int_0^x k (t,y) q(y) m(y, x) dy, 0< t< x$$ and the kernelk is $$k(t, y) = min(t, y)$$      

Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.     

О РАцИОНАльНых сОстА ВльУЩИх МЕРОМОРФНых ФУНкцИИ И Их пРОИжВОД Ных

LetG be an arbitrary domain in \(\bar C\) ,f a function meromorphic inG, $$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$ andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set $$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$ in case ∞?G, andα=0 in case ∞?G. It is proved that $$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$ where $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$      

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.     

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.     

A Geometry for the Set of Split Operators

We study the set ${\mathcal{X}}$ of split operators acting in the Hilbert space ${\mathcal{H}}$ : $$\mathcal{X}=\{T\in \mathcal{B}(\mathcal{H}): N(T)\cap R(T)=\{0\} \ {\rm and} \ N(T)+R(T)=\mathcal{H}\}.$$ Inside ${\mathcal{X}}$ , we consider the set ${\mathcal{Y}}$ : $$\mathcal{Y}=\{T\in\mathcal{X}: N(T)\perp R(T)\}.$$ Several characterizations of these sets are given. For instance ${T\in\mathcal{X}}$ if and only if there exists an oblique projection ${Q}$ whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. TS = ST, TST = T and STS = S). Analogous characterizations are given for ${\mathcal{Y}}$ . Two natural maps are considered: $${\bf q}:\mathcal{X} \to \mathbb{Q}:=\{{\rm oblique \ projections \ in} \, \mathcal{H} \}, \ {\bf q}(T)=P_{R(T)//N(T)}$$ and $${\bf p}:\mathcal{Y} \to \mathbb{P}:=\{{\rm orthogonal \ projections \ in} \ \mathcal{H} \}, \ {\bf p}(T)=P_{R(T)}, $$ where ${P_{R(T)//N(T)}}$ denotes the projection onto R(T) with nullspace N(T), and P _R(T) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets ${\mathcal{X}_{c_k}\subset \mathcal{X}}$ of operators with rank ${k<\infty}$ , and ${\mathcal{X}_{F_k}\subset\mathcal{X}}$ of Fredholm operators with nullity ${k<\infty}$ . For the map p there are analogous results. We show that the interior of ${\mathcal{X}}$ is ${\mathcal{X}_{F_0}\cup\mathcal{X}_{F_1}}$ , and that ${\mathcal{X}_{c_k}}$ and ${\mathcal{X}_{F_k}}$ are arc-wise connected differentiable manifolds.     

Comparison and oscillatory behavior for certain second order nonlinear dynamic equations

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)=0$$ on an arbitrary time scale $\mathbb{T}$ , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on $\mathbb{T}$ . Comparison results with the inequality $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)\leqslant 0\quad (\geqslant 0)$$ are established and application to neutral equations of the form $$\bigl(a(t)\bigl(\bigl[x(t)+p(t)x[\tau (t)]\bigr]^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }+q(t)x^{\beta }\bigl[g(t)\bigr]=0$$ are investigated.     

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

On semi-classical limits of ground states of a nonlinear Maxwell–Dirac system

We study the semi-classical ground states of the nonlinear Maxwell–Dirac system: $$\begin{aligned} \left\{ \begin{array}{l} \alpha \cdot \left( i\hbar \nabla + q(x)\mathbf{A }(x)\right) w-a\beta w -\omega w - q(x)\phi (x) w = P(x)g(\left| w\right| ) w\\ -\Delta \phi =q(x)\left| w\right| ^2\\ -\Delta {A_k}=q(x)(\alpha _k w)\cdot \bar{w}\ \ \ \ k=1,2,3 \end{array} \right. \end{aligned}$$ for \(x\in \mathbb{R }^3\) , where \(\mathbf{A }\) is the magnetic field, \(\phi \) is the electron field and \(q\) describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for \(\hbar \) small. We also describe the concentration behavior of the solutions as \(\hbar \rightarrow 0\) .     

A discreteness criterion for the spectrum of a quasielliptic operator

For the spectrum of the operator $$u = \sum\nolimits_{j = 1}^n {( - 1)^{m_j } D_j^{2m_j } u + q(x)u,} $$ to be discrete, where the m_j are arbitrary positive integers such that \(\sum\nolimits_{j = 1}^n {\tfrac{1}{{2m_j }}< 1} \) , and q(x) ≥ 1, it is necessary and sufficient that \(\int\limits_K {q (x) dx \to \infty } \) , when the cube K tends to infinity while preserving its dimensions.     

Bombieri's theorem in short intervals

The well-known Bombieri-A. I. Vinogradov theorem states that (1) $$\sum\limits_{q \leqslant x^{\tfrac{1}{2}} (\log x)^{ - s} } {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{y \leqslant x} } \left| {\psi (y,q;a) - \frac{y}{{\varphi (q)}}} \right| \ll \frac{x}{{(\log x)^A }},$$ whereA is an arbitrary positive constant,B=B(A)>0, and as usual, $$\psi (x,q;a) = \sum\limits_{\mathop {n \leqslant x}\limits_{n = a(q)} } {\Lambda (n),}$$ Λ being the Von Mangoldt's function. The problem of finding a result analogous to (1) for short intervals was investigated by many authors. Using Heath-Brown's identity and the approximate functional equation for DirichletL-functions, A. Perelli, J. Pintz and S. Salerno in 1985 established the following extension of Bombieri's theorem: Theorem 1. (2) $$\sum\limits_{q \leqslant Q} {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{h \leqslant y} \mathop {\max }\limits_{\frac{x}{2}< \approx \leqslant x} } \left| {\psi (z + h,q;a) - \psi (z,q;a) - \frac{h}{{\varphi (q)}}} \right| \ll \frac{y}{{(\log x)^A }}$$ where A>0 is an arbitrary constant,y=x ^θ $$\frac{7}{{12}}< \theta \leqslant 1, Q = x^{\frac{1}{{40}}} .$$ ,Q=x ^1/40. By improving the basic lemma which A. Perelli, J. Pintz and S. Salerno used as the main tool to prove Theorem 1, we obtain Theorem 2.Under the same condition as in Theorem 1,for Q=x ^1/38.5, (2)still holds.     

Optimal trajectories for quadratic variational processes via invariant imbedding

Consider minimizing the integral $$I = \int_0^T {[\dot w^2 + g(y)w^2 ] dy}$$ where $$w = w(y), \dot w = dw/dy, w(T) = 1, w(0) = free$$ ForT sufficiently small, it is shown that $$w_{opt} = x(t,T), 0 \leqslant t \leqslant T$$ where the functionx, viewed as a function ofT, is a solution of the Cauchy problem $$\begin{gathered} x_T (t,T) = r(T)x(t,T), T \geqslant t \hfill \\ x(t,t) = 1 \hfill \\\end{gathered}$$ and the auxiliary functionr satisfies the Riccati system $$\begin{gathered} r_T = ---g(T) + r^2 , T \geqslant 0 \hfill \\ r(0) = 0 \hfill \\\end{gathered}$$ In the derivation of the Cauchy problem, no use is made of Euler equations, dynamic programming, or Pontryagin's maximum principle. Only ordinary differential equations are employed. The Cauchy problem provides a one-sweep integration procedure; it is intimately connected with the theory of the second variation.