On Representations of Integers in Thin Subgroups of {{\rm SL}_2({\mathbb {Z}})}

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ ? \mathbb Z²  \  {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv₀g, w₀?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on \mathbb R²{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set \mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S   (mod  q)   for all   q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings, |\mathfrak E(N)| << N^1-e₀{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞.     

Local–global divisibility by 4 in elliptic curves defined over {\mathbb{Q}}

Let ${\mathcal{E}}Let E{\mathcal{E}} be an elliptic curve defined over \mathbbQ{\mathbb{Q}} . Let P ? E(\mathbb Q){P\in {\mathcal{E}}(\mathbb {Q})} and let q be a positive integer. Assume that for almost all valuations v ? \mathbbQ{v\in \mathbb{Q}} , there exist points D_v ? E(\mathbb Q_v){D_v\in {\mathcal{E}}(\mathbb {Q}_v)} such that P = qD _v . Is it possible to conclude that there exists a point D ? E(\mathbb Q){D\in {\mathcal{E}}(\mathbb {Q})} such that P = qD? A full answer to this question is known when q is a power of almost all primes p ? \mathbbN{p\in \mathbb{N}} , but some cases remain open when p ? S={2,3,5,7,11,13,17,19,37,43,67,163}{p\in S=\{2,3,5,7,11,13,17,19,37,43,67,163\}} . We now give a complete answer in the case when q = 4.     

All 4-Edge-Connected HHD-Free Graphs are {\mathbb{Z}_3}-Connected

An undirected graph G = (V, E) is called \mathbbZ₃{\mathbb{Z}_3}-connected if for all b: V ? \mathbbZ₃{b: V \rightarrow \mathbb{Z}_3} with ?_{v ? V}b(v)=0{\sum_{v \in V}b(v)=0}, an orientation D = (V, A) of G has a \mathbbZ₃{\mathbb{Z}_3}-valued nowhere-zero flow f: A? \mathbbZ₃-{0}{f: A\rightarrow \mathbb{Z}_3-\{0\}} such that ?_{e ? d⁺(v)}f(e)-?_{e ? d^-(v)}f(e)=b(v){\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)} for all v ? V{v \in V}. We show that all 4-edge-connected HHD-free graphs are \mathbbZ₃{\mathbb{Z}_3}-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the \mathbbZ₃{\mathbb{Z}_3}-connectivity for 4-edge-connected chordal graphs.     

A new construction of anti-self-dual four-manifolds

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D ₂ V = 0 on a five-dimensional manifold Y{\mathbf{Y}}. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in {V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space Z{\mathcal{Z}} has dim|-\frac12K_Z| 3 2{ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}, projecting thus over \mathbb C\mathbb P₂{\mathbb C\mathbb P_2} with twistor lines mapping onto plane conics.     

Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms

Let X, X ₁, X ₂,… be i.i.d. \mathbbR^d {\mathbb{R}^d} -valued real random vectors. Assume that E X = 0 and that X has a nondegenerate distribution. Let G be a mean zero Gaussian random vector with the same covariance operator as that of X. We study the distributions of nondegenerate quadratic forms \mathbbQ[ S_N ] \mathbb{Q}\left[ {{S_N}} \right] of the normalized sums S _N  = N ^−1/2 (X ₁ + ⋯ + X _N ) and show that, without any additional conditions,

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      7. $${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel      Cristina Fernández-Córdoba  Jaume Pujol  Mercè Villanueva 《Designs, Codes and Cryptography》2010,56(1):43-59 A code C{{\mathcal C}} is \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ2\mathbbZ4{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.      8. Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}      Gabriel Chiriacescu  Mihnea Colţoiu  Cezar Joiţa 《Mathematische Zeitschrift》2010,264(3):671-677 We show that if A is a closed analytic subset of \mathbbPn{\mathbb{P}^n} of pure codimension q then Hi(\mathbbPn\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that Hn-2(\mathbbPn\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .      9. Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation      Lawrence A. Fialkow 《Integral Equations and Operator Theory》2003,45(4):405-435 We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz 2 ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g(2n):g00,g01,g10,?,g0,2n,g1,2n-1,?,g2n-1,1,g2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that gij=ò[`(z)]izj dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ 2, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number gn,n+1 \gamma_{n,n+1} satisfying gn+1,n o [`(g)]n,n+1=Agn,n-1+Bgn,n+Cgn+1,n-1+Dgn,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.      10. Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips      D. Danielli  N. Garofalo  D. M. Nhieu 《Mathematische Zeitschrift》2010,265(3):617-637 We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H1{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H1{{\mathbb {H}}^1} the quantity \fracsH(S?B(p,r))rQ-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H1{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.      11. Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules      Patrice P. Ntumba 《Mediterranean Journal of Mathematics》2010,7(4):445-454 Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.      12. On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$      Christophe Dupont 《Mathematische Annalen》2011,349(3):509-528 Let f be an endomorphism of \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ 1, . . . , λ k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dimHm 3 [(log d)/(l1)] + [(log d)/(l2)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f n in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f n .      13. Embeddings of general curves in projective spaces: the range of the quadrics      Edoardo Ballico 《Lithuanian Mathematical Journal》2012,52(2):134-137 Let C ì \mathbbPr C \subset {\mathbb{P}^r} be a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either h0( \mathbbPr,IC(2) ) = 0 {h^0}\left( {{\mathbb{P}^r},{\mathcal{I}_C}(2)} \right) = 0 or h1( \mathbbPr,IC(2) ) = 0 {h^1}\left( {{\mathbb{P}^r},{\mathcal{I}_C}(2)} \right) = 0 (a problem called the maximal rank conjecture in the range of quadrics).      14. Maximal Chains in Positive Subfamilies of P(ω)      Milo? S. Kurili? 《Order》2012,29(1):119-129 A family P ì [w]w{\mathcal P} \subset [\omega]^\omega is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/Fin. For example, if I ì P(w){\mathcal I} \subset P(\omega) is an ideal containing the ideal Fin of finite subsets of ω, then P(w) \IP(\omega) \setminus {\mathcal I} is a positive family and the set Dense(\mathbb Q)\mbox{Dense}({\mathbb Q}) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(\mathbb Q)P({\mathbb Q}). We prove that, for a positive family P{\mathcal P}, the order types of maximal chains in the complete lattice áP è{?}, ì ?\langle {\mathcal P} \cup \{\emptyset\}, \subset \rangle are exactly the order types of compact nowhere dense subsets of the real line having the minimum non-isolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0,1)\mathbb R\mbox{Intalg}[0,1)_{{\mathbb R}} and the poset E(\mathbb Q)E({\mathbb Q}), where E(\mathbb Q)E({\mathbb Q}) is the set of elementary submodels of the rational line.      15. Singularities of secant maps of immersed surfaces      Sunayana Ghosh  Joachim H. Rieger 《Geometriae Dedicata》2006,121(1):73-87 The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion $X\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c}The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion X:\mathbb Rn ? \mathbb Rn+cX\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c} at the diagonal in the source is a \mathbb Z2{\mathbb Z}_2 stable map-germ \mathbb R2n ? \mathbb Rn+c-1{\mathbb R}^{2n} \to {\mathbb R}^{n+c-1} in the following cases: (i) c≥ 2 and (2n,n + c − 1) is a pair of dimensions for which the \mathbb Z2{\mathbb Z}_2 stable germs of rank at least n are dense, and (ii) for generically immersed surfaces (i.e., n = 2 and any c≥ 1). In the latter surface case the A\mathbb Z2{\mathcal A}^{{\mathbb Z}_2}-classification of germs of secant maps at the diagonal is described and it is related to the A{\mathcal A}-classification of certain singular projections of the surfaces.      16. On the existence of a p-adic metaplectic Tate-type $\widetilde {\gamma}$-factor      Dani Szpruch 《The Ramanujan Journal》2011,26(1):45-53 Let \mathbbF\mathbb{F} be a p-adic field, let χ be a character of \mathbbF*\mathbb{F}^{*}, let ψ be a character of \mathbbF\mathbb{F} and let gy-1\gamma_{\psi}^{-1} be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic function [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi ,s,\psi) via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against y·gy-1\psi\cdot\gamma_{\psi}^{-1}. It turns out that γ and [(g)\tilde]\widetilde{\gamma} have similar integral representations. Furthermore, [(g)\tilde]\widetilde{\gamma} has a relation to Shahidi‘s metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi‘s local coefficient. Up to an exponential factor, [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi,s,\psi) is equal to the ratio \fracg(c2,2s,y)g(c,s+\frac12,y)\frac{\gamma(\chi^{2},2s,\psi)}{\gamma(\chi,s+\frac{1}{2},\psi)}.      17. On the singularity of the irreducible components of a Springer fiber in {\mathfrak{s}\mathfrak{l}_n}      Lucas Fresse  Anna Melnikov 《Selecta Mathematica, New Series》2010,16(3):393-418 Let ${\mathcal{B}_u}Let Bu{\mathcal{B}_u} be the Springer fiber over a nilpotent endomorphism u ? End(\mathbbCn){u\in {\rm End}(\mathbb{C}^n)}. Let J (u) be the Jordan form of u regarded as a partition of n. The irreducible components of Bu{\mathcal{B}_u} are all of the same dimension. They are labelled by Young tableaux of shape J (u). We study the question of the singularity of the components of Bu{\mathcal{B}_u} and show that all the components of Bu{\mathcal{B}_u} are nonsingular if and only if J(u) ? {(l,1,1,?), (l1,l2), (l1,l2,1), (2,2,2)}{J(u)\in\{(\lambda,1,1,\ldots), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}}.      18. Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}      Mohamed Benrhouma  Hichem Ounaies 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(5):647-662 In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem {l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      19. The Ground State and the Long-Time Evolution in the CMC Einstein Flow      Martin Reiris 《Annales Henri Poincare》2010,10(8):1559-1604 Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)3Volg(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by Vinf=(\frac-16Y(S))\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g i , K i )} satisfying: (i) k i  = −3, (ii) Viˉ Vinf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q 0((g i , K i )) ≤ Λ, where Q 0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)2g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E1)\tilde]=E1(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E1=Q0+Q1{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ Vinf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.      20. Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$      Frédéric A. B. Edoukou 《Designs, Codes and Cryptography》2009,50(1):135-146 We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .

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$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}

We show that if A is a closed analytic subset of \mathbbPⁿ{\mathbb{P}^n} of pure codimension q then Hⁱ(\mathbbPⁿ\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that H^n-2(\mathbbPⁿ\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .     

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.     

Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H¹{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H¹{{\mathbb {H}}^1} the quantity \fracs_H(S?B(p,r))r^Q-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H¹{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.     

Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.     

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

Embeddings of general curves in projective spaces: the range of the quadrics

Let C ì \mathbbP^r C \subset {\mathbb{P}^r} be a general embedding of prescribed degree of a general smooth curve with prescribed genus. Here we prove that either h⁰( \mathbbP^r,I_C(2) ) = 0 {h^0}\left( {{\mathbb{P}^r},{\mathcal{I}_C}(2)} \right) = 0 or h¹( \mathbbP^r,I_C(2) ) = 0 {h^1}\left( {{\mathbb{P}^r},{\mathcal{I}_C}(2)} \right) = 0 (a problem called the maximal rank conjecture in the range of quadrics).     

Maximal Chains in Positive Subfamilies of P(ω)

A family P ì [w]^w{\mathcal P} \subset [\omega]^\omega is called positive iff it is the union of some infinite upper set in the Boolean algebra P(ω)/Fin. For example, if I ì P(w){\mathcal I} \subset P(\omega) is an ideal containing the ideal Fin of finite subsets of ω, then P(w) \IP(\omega) \setminus {\mathcal I} is a positive family and the set Dense(\mathbb Q)\mbox{Dense}({\mathbb Q}) of dense subsets of the rational line is a positive family which is not the complement of some ideal on P(\mathbb Q)P({\mathbb Q}). We prove that, for a positive family P{\mathcal P}, the order types of maximal chains in the complete lattice áP è{?}, ì ?\langle {\mathcal P} \cup \{\emptyset\}, \subset \rangle are exactly the order types of compact nowhere dense subsets of the real line having the minimum non-isolated. Also we compare this result with the corresponding results concerning maximal chains in the Boolean algebras P(ω) and Intalg[0,1)_{\mathbb R}\mbox{Intalg}[0,1)_{{\mathbb R}} and the poset E(\mathbb Q)E({\mathbb Q}), where E(\mathbb Q)E({\mathbb Q}) is the set of elementary submodels of the rational line.     

Singularities of secant maps of immersed surfaces

The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion $X\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c}The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion X:\mathbb Rⁿ ? \mathbb R^n+cX\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c} at the diagonal in the source is a \mathbb Z₂{\mathbb Z}_2 stable map-germ \mathbb R²ⁿ ? \mathbb R^n+c-1{\mathbb R}^{2n} \to {\mathbb R}^{n+c-1} in the following cases: (i) c≥ 2 and (2n,n + c − 1) is a pair of dimensions for which the \mathbb Z₂{\mathbb Z}_2 stable germs of rank at least n are dense, and (ii) for generically immersed surfaces (i.e., n = 2 and any c≥ 1). In the latter surface case the A^{\mathbb Z₂}{\mathcal A}^{{\mathbb Z}_2}-classification of germs of secant maps at the diagonal is described and it is related to the A{\mathcal A}-classification of certain singular projections of the surfaces.     

On the existence of a p-adic metaplectic Tate-type $\widetilde {\gamma}$-factor

Let \mathbbF\mathbb{F} be a p-adic field, let χ be a character of \mathbbF^*\mathbb{F}^{*}, let ψ be a character of \mathbbF\mathbb{F} and let g_y^-1\gamma_{\psi}^{-1} be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic function [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi ,s,\psi) via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against y·g_y^-1\psi\cdot\gamma_{\psi}^{-1}. It turns out that γ and [(g)\tilde]\widetilde{\gamma} have similar integral representations. Furthermore, [(g)\tilde]\widetilde{\gamma} has a relation to Shahidi‘s metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi‘s local coefficient. Up to an exponential factor, [(g)\tilde](c,s,y)\widetilde{\gamma}(\chi,s,\psi) is equal to the ratio \fracg(c²,2s,y)g(c,s+\frac12,y)\frac{\gamma(\chi^{2},2s,\psi)}{\gamma(\chi,s+\frac{1}{2},\psi)}.     

On the singularity of the irreducible components of a Springer fiber in {\mathfrak{s}\mathfrak{l}_n}

Let ${\mathcal{B}_u}Let B_u{\mathcal{B}_u} be the Springer fiber over a nilpotent endomorphism u ? End(\mathbbCⁿ){u\in {\rm End}(\mathbb{C}^n)}. Let J (u) be the Jordan form of u regarded as a partition of n. The irreducible components of B_u{\mathcal{B}_u} are all of the same dimension. They are labelled by Young tableaux of shape J (u). We study the question of the singularity of the components of B_u{\mathcal{B}_u} and show that all the components of B_u{\mathcal{B}_u} are nonsingular if and only if J(u) ? {(l,1,1,?), (l₁,l₂), (l₁,l₂,1), (2,2,2)}{J(u)\in\{(\lambda,1,1,\ldots), (\lambda_1,\lambda_2), (\lambda_1,\lambda_2,1), (2,2,2)\}}.     

Existence of solutions for a perturbation sublinear elliptic equation in {\mathbb {R}^N}

In this paper we consider the following problem $\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.$ ${f \in L^2(\mathbb{R}^N)\cap L^\frac{2(1-\theta)}{1-2\theta}(\mathbb{R}^N),\, N\geq 3,\, f\geq 0,\, f \neq 0}In this paper we consider the following problem

{l -Du=u-|u|-2qu+f u ? H1(\mathbbRN)?L2(1-q)(\mathbbRN)\left\{\begin{array}{l} -\Delta u=u-\left|u\right|^{-2\theta}u+f \\u \in H^1(\mathbb{R}^N)\cap L^{2(1-\theta)}(\mathbb{R}^N)\end{array}\right.      19. The Ground State and the Long-Time Evolution in the CMC Einstein Flow      Martin Reiris 《Annales Henri Poincare》2010,10(8):1559-1604 Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)3Volg(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by Vinf=(\frac-16Y(S))\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g i , K i )} satisfying: (i) k i  = −3, (ii) Viˉ Vinf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q 0((g i , K i )) ≤ Λ, where Q 0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)2g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E1)\tilde]=E1(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E1=Q0+Q1{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ Vinf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.      20. Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$      Frédéric A. B. Edoukou 《Designs, Codes and Cryptography》2009,50(1):135-146 We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

Codes defined by forms of degree 2 on non-degenerate Hermitian varieties in $${\mathbb{P}^{4}(\mathbb{F}_q)}$$

We study the functional codes of second order on a non-degenerate Hermitian variety as defined by G. Lachaud. We provide the best possible bounds for the number of points of quadratic sections of . We list the first five weights, describe the corresponding codewords and compute their number. The paper ends with two conjectures. The first is about minimum distance of the functional codes of order h on a non-singular Hermitian variety . The second is about distribution of the codewords of first five weights of the functional codes of second order on a non-singular Hermitian variety .