Precise inequalities for norms of functions,third partial,second mixed,or directional derivatives

For functions f which are bounded throughout the plane R₂ together with the partial derivatives f(^3,0) f(_0,3), inequalities $$\left\| {f^{(1,1)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} ,\left\| {f_e^{(2)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left( {\left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_1 } \right| + \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_2 } \right|} \right)^2 ,$$ are established, where ∥?∥denotes the upper bound on R² of the absolute values of the corresponding function, andf _fe ⁽²⁾ is the second derivative in the direction of the unit vector e=(e₁, e₂). Functions are exhibited for which these inequalities become equalities.     

Regular orthoscalar representations of the extended Dynkin graph $ {\tilde{E}_8} $ AND ∗-algebra associatedwith it

We obtain a classification of regular orthoscalar representations of the extended Dynkin graph [(E)\tilde]₈ {\tilde{E}_8} with special character. Using this classification, we describe triples of self-adjoint operators A, B, and C such that their spectra are contained in the sets {0, 1, 2, 3, 4, 5}, {0, 2, 4}, and {0, 3}, respectively, and the equality A + B + C = 6I is true.     

Some Topological Properties of Solution Sets in a Second Order Differential Inclusion with m-point Boundary Conditions

We consider a class of m-point (m > 3) second order boundary value problem (P_F\cal{P}_{F}) in a separable Banach space E of the form W ^{2, 1}_E([0, 1])W ^{2, 1}_E([0, 1])-solutions of (P_F\cal{P}_{F}) is compact and is a retract in C¹_E([0, 1])C^1_E([0, 1]). A new existence result of W ^{2, 1}_E([0, 1])W ^{2, 1}_E([0, 1])-solutions and a related relaxation problem are also provided, here F is no longer bounded.     

Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic formsQ X _j X _k+ (X _j ² -E X _j ² )where X _j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a _jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.     

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

Duality for Harmonic Differential Forms Via Clifford Analysis

The space HF _k (Ω) of harmonic multi-vector fields in a domain as introduced in [1] is closely connected to the space of harmonic forms. The main aim of this paper is to characterize the dual space of HF _k (E) being a compact set. It is proved that HF _k (E)^* is isomorphic to a certain quotient space of so-called harmonic pairs outside E vanishing at infinity. Research of the third author was supported by the FWO Research Network WO. 003. 01N, research of the fourth author was supported by the FWO “Krediet aan Navorsers: 1.5.106.02”     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

Relative Artin motives and the reductive Borel–Serre compactification of a locally symmetric variety

We introduce the notion of Artin motives and cohomological motives over a scheme X. Given a cohomological motive M over X, we consider the universal Artin motive mapping to M and denote it w⁰_X(M)\omega^{0}_{X}(M). We use this to define a motive \mathbbE_X\mathbb{E}_{X} over X which is an invariant of the singularities of X. The first half of the paper is devoted to the study of the functors w⁰_X\omega^{0}_{X} and the computation of the motives \mathbbE_X\mathbb{E}_{X}.     

Liftings for Haar measure on {0,1} k

We callE ⊆ {0,1} ^k projective if for some countableA ⊆κ there is anE _A ⊆ {0, 1} ^A such thatE=E _A ×{0,1} ^k\A andE _A is a projective subset of the Cantor set {0, 1} ^A . We construct a model where Haar measure on {0,1} ^k has no projective lifting (and in particular no Baire lifting) for anyκ≥ω. Research partially supported by NATO Science Fellowship. The first author would like to thank the Mathematics Department at the University of Essex for its hospitality during the academic year 1988/89 while part of this research was being carried out. This research was initiated while the second author was a postdoctoral fellow at the University of Toronto. Its completion was supported by NSF grant DMS-8505550.     

Projective limits of finite decomposition systems

Special finite topological decomposition systems were used to get compactifications of topological spaces in [6]. In this paper the notion of finite decomposition systems is applied for topological measure spaces. We get two canonical topological measure spaces X_∞ and X_∞^d being projective limits of (discrete) finite decomposition systems for each topological measure space X = (X, O, A, P) and each net (A_α) α ? I of upward filtering finite σ-algebras in A. X_∞ is a compact topological measure space and the idea to construct is the same as used in [6]. The compactifications of [6] are cases of some special X_∞. Further on we obtain that each measurable set of the remainder of X_∞ has measure zero with respect to the limit measure P_∞ (Theorem 1). X_∞^d is the STONE representation space X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document} A_α, hence a Boolean measure space with regular Borel measure. Some measure theoretical and topological relations between X, X(\documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \cup \limits_{\alpha \in I} A\alpha $\end{document}) and x(A) where x(A) is the Stone representation space of A, are given in Theorem 2. and 4. As a corollary from Theorem 2. we get a measure theoretical-topological version to the Theorem of Alexandroff Hausdorff for compact T₂ measure spaces x with regular Borel measure (Theorem 3.).     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , or [(E)\tilde]₇ {\tilde E_7} . An algebra TL_{G, t} T{L_{\Gamma, \tau }} has exponential growth if and only if the graph Γ coincides with none of the graphs A_n {A_n} , D_n {D_n} , E_n {E_n} , [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , and [(E)\tilde]₇ {\tilde E_7} .     

Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs

Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C^*-algebra C^*(E) of E which is generated by a Cuntz-Krieger E-family {s_e, p_v}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C^*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.     

Convex Embeddings and Bisections of 3-Connected Graphs1

Given two disjoint subsets T ₁ and T ₂ of nodes in an undirected 3-connected graph G = (V, E) with node set V and arc set E, where and are even numbers, we show that V can be partitioned into two sets V ₁ and V ₂ such that the graphs induced by V ₁ and V ₂ are both connected and holds for each j = 1,2. Such a partition can be found in time. Our proof relies on geometric arguments. We define a new type of convex embedding of k-connected graphs into real space R ^k-1 and prove that for k = 3 such an embedding always exists. ¹ A preliminary version of this paper with title Bisecting Two Subsets in 3-Connected Graphs appeared in the Proceedings of the 10th Annual International Symposium on Algorithms and Computation, ISAAC 99, (A. Aggarwal, C. P. Rangan, eds.), Springer LNCS 1741, 425–434, 1999.     

Linear maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

Generators of the unoriented bordism ring which are fibered over products of projective spacesRP(2 r −1)

For each positive integerk≦∞ we construct a family {M _k ⁿ } of generators of the unoriented bordims ring. The manifoldsM _k ⁿ are total spaces of fiber bundles whose base spaces are high-dimensional products of projective spaces wherer _i≦k. The fibers are themselves iterated projective bundles with maximal fiber dimension two. In the special casek=3 we obtain generatorsM ₃ ⁿ which admit approximately 7/8·n pointwise linearly independent vector fields. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

On the error terms for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant −D, and r _f (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $ \sum\limits_{n \leqq x} {r_f (n)^\beta } $ \sum\limits_{n \leqq x} {r_f (n)^\beta } for any positive integer β. Next, we give an estimate of ∫₁ ^T |E(x)|² x ^−3/2 dx when β = 1.     

On some exotic finite subgroups of E 8 and Springer’s regular elements of the Weyl group

A proof (by Serre and by Cohen, Griess and Lisser) verified, in the special case of E ₈, a conjecture of mine, that the finite projective group L ₂(61) embeds in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) . Subsequently, Griess and Ryba have shown (using computers) that L ₂(49) and, in addition, (established by Serre without computers) L ₂(41) also embed in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) . That is, if K = 30, 24, 20 and k ∈ K then L ₂(2k + 1) embeds in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) . In this paper we show that the “Borel” subgroup B(k) of L ₂(2k + 1), k ∈ K, has a uniform construction. The theorem uses a result of T. Springer on the existence in E₈( \mathbbC ) {E_8}\left( \mathbb{C} \right) of three regular elements of the Weyl group, having orders k ∈ K, and associated to the regular, subregular and subsubregular nilpotent elements. Springer’s result generalizes (in the E ₈ case) a 1959 general result of mine relating the principal nilpotent element with the Coxeter element.     

Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices H in d \geqslant 1{d \geqslant 1} dimensions. The matrix elements H _xy , indexed by x,y ? L ì \mathbbZ^d{x,y \in \Lambda \subset \mathbb{Z}^d}, are independent and their variances satisfy s_xy²:=\mathbbE |H_xy|² = W^-d f((x - y)/W){\sigma_{xy}^2:=\mathbb{E} |{H_{xy}}|^2 = W^{-d} f((x - y)/W)} for some probability density f. We assume that the law of each matrix element H _xy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t << W^d/3{t\ll W^{d/3}} . We also show that the localization length of the eigenvectors of H is larger than a factor W^d/6{W^{d/6}} times the band width W. All results are uniform in the size |Λ| of the matrix. This extends our recent result (Erdős and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying ?_xs_xy²=1{\sum_x\sigma_{xy}^2=1} for all y, the largest eigenvalue of H is bounded with high probability by 2 + M^{-2/3 + e}{2 + M^{-2/3 + \varepsilon}} for any ${\varepsilon > 0}${\varepsilon > 0}, where M : = 1 / (max_x,ys_xy²){M := 1 / (\max_{x,y}\sigma_{xy}^2)} .     

Empirical Processes with a Bounded Ψ 1 Diameter

We study the empirical process ${{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}We study the empirical process sup_{f ? F}|N^-1?_i=1^N f²(X_i)-\mathbbEf²|{{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}, where F is a class of mean-zero functions on a probability space (Ω, μ), and (X_i)_{i = 1}^N{(X_{i})_{i =1}^N} are selected independently according to μ.