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We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of $f$, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of $f$ on $\mathcal O\times\mathcal O$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the $\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the real hyperbolic $5$-space., Comment: Revised version, 40 pages

We construct a family of Hermitian metrics on the Hopf surface $ \mathbb{S}^3\times \mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally K\"ahler. Among the toric fibres of $\pi:\mathbb{S}^{3} \times \mathbb{S}^1\to\mathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces., Comment: We generalize Theorem 1.2 to $\mathbb{S}^{2n-1}\times\mathbb{S}^1$ in section 4

We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.

If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{eigenvector-eigenvalue identity} and show how this identity can also be used to extract the relative phases between the components of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity., Comment: 28 pages, 1 figure, phase discussion expanded, matches published version

In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of F q 2 \mathbb {F}_{q^2} -rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q . d \le q.

Our goal is to combine the techniques of Xiaokui Yang, Valentino Tosatti, and others to establish a Liouville-type result for almost complex manifolds. The transition to the non-integrable setting is delicate, so we will devote a section to discuss the key differences, and another to introduce the tools we will be using. Afterwards, we present a proof of our main theorem.

We explore the regularity of the roots of Garding hyperbolic polynomials and real stable polynomials. As an application we obtain new regularity results of Sobolev type for the eigenvalues of Hermitian matrices and for the singular values of arbitrary matrices. These results are optimal among all Sobolev spaces., 13 pages; some details added and typos corrected, accepted for publication in Proc. Amer. Math. Soc

We construct a lifting that associates to a Hilbert cusp form a Hilbert-Hermitian cusp form. This is a generalization of the lifting of elliptic cusp forms constructed by Ikeda to arbitrary Hilbert cusp forms.

Given a collection λ _ = { λ 1 \underline {\lambda }=\{\lambda _1 , … \dots , λ n } \lambda _n\} of real numbers, there is a canonical probability distribution on the set of real symmetric or complex Hermitian matrices with eigenvalues λ 1 , … , λ n \lambda _1, \ldots , \lambda _n . In this paper, we study various features of random matrices with this distribution. Our main results show that under mild conditions, when n n is large, linear functionals of the entries of such random matrices have approximately Gaussian joint distributions. The results take the form of upper bounds on distances between multivariate distributions, which allows us also to consider the case when the number of linear functionals grows with n n . In the context of quantum mechanics, these results can be viewed as describing the joint probability distribution of the expectation values of a family of observables on a quantum system in a random mixed state. Other applications are given to spectral distributions of submatrices, the classical invariant ensembles, and to a probabilistic counterpart of the Schur–Horn theorem, relating eigenvalues and diagonal entries of Hermitian matrices.

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $sl(2,\mathbb{C})$, generalizing the well known structure on the harmonic forms of compact K\"ahler manifolds. Some topological implications are deduced., Comment: 7 pages, to appear in Proc. AMS

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-K\"ahler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in \cite{EFV} for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures., Comment: 25 pages. Revised version. To appear in TAMS

We give several versions of Siu’s ∂ ∂ ¯ \partial \overline {\partial } -formula for maps from a strictly pseudoconvex pseudo-Hermitian manifold ( M 2 m + 1 , θ ) (M^{2m+1}, \theta ) into a Kähler manifold ( N n , g ) (N^n, g) . We also define and study the notion of pseudo-Hermitian harmonicity for maps from M M into N N . In particular, we prove a CR version of the Siu Rigidity Theorem for pseudo-Hermitian harmonic maps from a pseudo-Hermitian manifold with vanishing Webster torsion into a Kähler manifold having strongly negative curvature.

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corresponding eigenvalues and we use these estimates to deduce that the associated heat operators are trace-class. Finally we give various applications to the Hodge-Dolbeault operator and to the Hodge-Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension $2$., To appear on Transactions of the American Mathematical Society. Comments are welcome. arXiv admin note: text overlap with arXiv:1607.00289

For each positive integer n n , let g Z ( n ) g_\mathbb {Z}(n) be the smallest integer such that if an integral quadratic form in n n variables can be written as a sum of squares of integral linear forms, then it can be written as a sum of g Z ( n ) g_\mathbb {Z}(n) squares of integral linear forms. We show that every positive definite integral quadratic form is equivalent to what we call a balanced Hermite–Korkin–Zolotarev-reduced form and use it to show that the growth of g Z ( n ) g_\mathbb {Z}(n) is at most an exponential of n \sqrt {n} . Our result improves the best known upper bound on g Z ( n ) g_\mathbb {Z}(n) which is on the order of an exponential of n n . We also define an analogous number g O ∗ ( n ) g_{\mathcal O}^*(n) for writing Hermitian forms over the ring of integers O \mathcal O of an imaginary quadratic field as sums of norms of integral linear forms, and when the class number of the imaginary quadratic field is 1 1 , we show that the growth of g O ∗ ( n ) g_{\mathcal O}^*(n) is at most an exponential of n \sqrt {n} . We also improve on results of both Conway and Sloane and Kim and Oh on s s -integrable lattices.

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-K\"ahler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.

Let p p be an odd prime. Let F F be the function field of a p p -adic curve. Let A A be a central simple algebra of period 2 over F F with an involution σ \sigma . There are known upper bounds for the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) . In this article we compute the exact values of the u u -invariant of hermitian forms over ( A , σ ) (A, \sigma ) .

We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchm\'aros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchm\'aros curve is equal to the above-mentioned ray class field extension of the Hermitian curve., Comment: 15 pages; various typos fixed

We study a class of Hermitian random matrices which includes and generalizes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as the adjacency matrices of Erdos-Renyi random graphs with p ~ 1/N. Our NxN random matrices have real entries which are i.i.d. up to symmetry. The distributions may depend on N, however, the sums of rows must converge in distribution; it is then well-known that the limiting distributions are infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure at the root of a Poisson weighted infinite skeleton tree. This graph is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure. One example covered by the results are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is just the positive integer line rooted at 1, and as expected, the LSD is Wigner's semi-circle law. The results also extend to self-adjoint complex matrices and also to Wishart matrices., Corrected a remark following Thm 1.3, Lemma 5.2, and a comment in Section 5. Various other minor revisions. 27 pages, 6 figures

In this paper, we introduce the first Aeppli-Chern class for complex manifolds and show that the ( 1 , 1 ) (1,1) -component of the curvature 2 2 -form of the Levi-Civita connection on the anti-canonical line bundle represents this class. We systematically investigate the relationship between a variety of Ricci curvatures on Hermitian manifolds and the background Riemannian manifolds. Moreover, we study non-Kähler Calabi-Yau manifolds by using the first Aeppli-Chern class and the Levi-Civita Ricci-flat metrics. In particular, we construct explicit Levi-Civita Ricci-flat metrics on Hopf manifolds S 2 n − 1 × S 1 \mathbb {S}^{2n-1}\times \mathbb {S}^1 . We also construct a smooth family of Gauduchon metrics on a compact Hermitian manifold such that the metrics are in the same first Aeppli-Chern class, and their first Chern-Ricci curvatures are the same and non-negative, but their Riemannian scalar curvatures are constant and vary smoothly between negative infinity and a positive number. In particular, it shows that Hermitian manifolds with non-negative first Chern class can admit Hermitian metrics with strictly negative Riemannian scalar curvature.

Given a sparse Hermitian matrix $A$ and a real number $\mu$, we construct a set of sparse vectors, each approximately spanned only by eigenvectors of $A$ corresponding to eigenvalues near $\mu$. This set of vectors spans the column space of a localized spectrum slicing (LSS) operator, and is called an LSS basis set. The sparsity of the LSS basis set is related to the decay properties of matrix Gaussian functions. We present a divide-and-conquer strategy with controllable error to construct the LSS basis set. This is a purely algebraic process using only submatrices of $A$, and can therefore be applied to general sparse Hermitian matrices. The LSS basis set leads to sparse projected matrices with reduced sizes, which allows the projected problems to be solved efficiently with techniques using sparse linear algebra. As an example, we demonstrate that the LSS basis set can be used to solve interior eigenvalue problems for a discretized second order partial differential operator in one-dimensional and two-dimensional domains, as well as for a matrix of general sparsity pattern.

Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T ( λ ) v = 0 T(\lambda )v=0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG method, including its global and asymptotic convergence. We propose several variants of single-vector and block PCG methods with deflation for computing multiple eigenvalues, and compare them in arithmetic and memory cost. Variable indefinite preconditioning is shown to be effective to accelerate convergence when some desired eigenvalues are not close to the lowest or highest eigenvalue. The efficiency of variants of PCG is illustrated by numerical experiments. Overall, the locally optimal block preconditioned conjugate gradient (LOBPCG) is the most efficient method, as in the linear setting.

We introduce the projective Hermite constant for positive definite binary hermitian forms associated with an imaginary quadratic number field K. It is a lower bound for the classical Hermite constant, and these two constants coincide when K has class number one. Using the geometric tools developed by Mendoza and Vogtmann for their study of the homology of the Bianchi groups, we compute the projective Hermite constants for those K whose absolute discriminants are less than 70, and determine the hermitian forms that attain the projective Hermite constants in these cases. A comparison of the projective hermitian constant with some other generalizations of the classical Hermite constant is also given. Fil: Chan, Wai Kiu. Wesleyan University; Estados Unidos Fil: Icaza, María Inés. Universidad de Talca; Chile Fil: Lauret, Emilio Agustin. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina

Topological quantum field theory associates to a punctured surface Σ \Sigma , a level r r and colors c c in { 1 , … , r − 1 } \{1,\ldots ,r-1\} at the marked points a finite dimensional Hermitian space V r ( Σ , c ) V_r(\Sigma ,c) . Curves γ \gamma on Σ \Sigma act as Hermitian operator T r γ T_r^\gamma on these spaces. In the case of the punctured torus and the 4-times punctured sphere, we prove that the matrix elements of T r γ T_r^\gamma have an asymptotic expansion in powers of 1 r \frac {1}{r} and we identify the two first terms using trace functions on representation spaces of the surface in S U 2 \mathrm {SU}_2 . We conjecture a formula for the general case. Then we show that the curve operators are Toeplitz operators on the sphere in the sense that T r γ = Π r f r γ Π r T_r^{\gamma }=\Pi _r f^\gamma _r\Pi _r where Π r \Pi _r is the Toeplitz projector and f r γ f^\gamma _r is an explicit function on the sphere which is smooth away from the poles. Using this formula, we show that under some assumptions on the colors associated to the marked points, the sequence T r γ T^\gamma _r is a Toeplitz operator in the usual sense with principal symbol equal to the trace function and with subleading term explicitly computed. We use this result and semi-classical analysis in order to compute the asymptotics of matrix elements of the representation of the mapping class group of Σ \Sigma on V r ( Σ , c ) V_r(\Sigma ,c) . We recover in this way the result of Taylor and Woodward on the asymptotics of the quantum 6j-symbols and treat the case of the punctured S-matrix.

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification of one K-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245-4261] in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of [D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, arXiv:1006.3822]), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.

Let R ⟨ x ⟩ \mathbb R\langle x \rangle denote the ring of polynomials in g g freely noncommuting variables x = ( x 1 , … , x g ) x=(x_1,\dots ,x_g) . There is a natural involution ∗ * on R ⟨ x ⟩ \mathbb R\langle x \rangle determined by x j ∗ = x j x_j^*=x_j and ( p q ) ∗ = q ∗ p ∗ (pq)^*=q^* p^* , and a free polynomial p ∈ R ⟨ x ⟩ p\in \mathbb R\langle x \rangle is symmetric if it is invariant under this involution. If X = ( X 1 , … , X g ) X=(X_1,\dots ,X_g) is a g g tuple of symmetric n × n n\times n matrices, then the evaluation p ( X ) p(X) is naturally defined and further p ∗ ( X ) = p ( X ) ∗ p^*(X)=p(X)^* . In particular, if p p is symmetric, then p ( X ) ∗ = p ( X ) p(X)^*=p(X) . The main result of this article says if p p is symmetric, p ( 0 ) = 0 p(0)=0 and for each n n and each symmetric positive definite n × n n\times n matrix A A the set { X : A − p ( X ) ≻ 0 } \{X:A-p(X)\succ 0\} is convex, then p p has degree at most two and is itself convex, or − p -p is a hermitian sum of squares.

It is shown that, up to scaling and isometry, the only complete 4-manifold with an Einstein metric of positive sectional curvature which is also Hermitian with respect to some complex structure is the complex projective plane C P 2 \mathbb {CP}_2 , equipped with its Fubini-Study metric.

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n then the corresponding modular variety is uniruled. We also construct new reflective modular forms and thus provide new examples of uniruled moduli spaces of lattice polarised K3 surfaces. Finally we prove that the moduli space of Kummer surfaces associated to (1,21)-polarised abelian surfaces is uniruled., Comment: 14 pages

Let M M be a Riemannian manifold and let E → M E\to M be a Hermitian vector bundle with a Hermitian covariant derivative ∇ \nabla . Furthermore, let H ( 0 ) H(0) denote the Friedrichs extension of ∇ ∗ ∇ / 2 \nabla ^*\nabla /2 and let V : M → E n d ( E ) V:M\to \mathrm {End}(E) be a potential. We prove that if V V has a decomposition of the form V = V 1 − V 2 V=V_1-V_2 with V j ≥ 0 V_j\geq 0 , V 1 V_1 locally integrable and | V 2 | \left | V_2 \right | in the Kato class of M M , then one can define the form sum H ( V ) := H ( 0 ) ∔ V H(V):=H(0)\dotplus V in Γ L 2 ( M , E ) \Gamma _{\mathsf {L}^2}(M,E) without any further assumptions on M M . Applications to quantum physics are discussed.

In this paper, we study the Nakano-positivity and dual-Nakano- positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if E E is an ample vector bundle over a compact Kähler manifold X X , S k E ⊗ det E S^kE\otimes \det E is both Nakano-positive and dual-Nakano-positive for any k ≥ 0 k\geq 0 . Moreover, H n , q ( X , S k E ⊗ det E ) = H q , n ( X , S k E ⊗ det E ) = 0 H^{n,q}(X,S^kE\otimes \det E)=H^{q,n}(X,S^kE\otimes \det E)=0 for any q ≥ 1 q\geq 1 . In particular, if ( E , h ) (E,h) is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle ( S k E ⊗ det E , S k h ⊗ det h ) (S^kE\otimes \det E, S^kh\otimes \det h) is both Nakano-positive and dual-Nakano-positive for any k ≥ 0 k\geq 0 .

We present a new proof and extension of the classical Sylvester Inertia Theorem to a pair of non-Hermitian matrices which satisfies the property that any real linear combination of the pair has only real eigenvalues. In the proof, we embed the given problem in a one-parameter family of related problems and examine the eigencurves of the family. The proof requires only elementary matrix theory and the Intermediate Value Theorem. The same technique is then used to extend Johnson’s extension of Sylvester’s Theorem on possible values of the inertia of a product of two matrices.

We consider the quadratic polynomials in three unitary generators, i.e. the elements of the group ∗ * -algebra of the free group with generators u 1 , u 2 , u 3 u_1, u_2, u_3 of the form f = ∑ j , k = 1 3 α j k u j ∗ u k f=\sum _{j, k=1}^{3}\alpha _{jk}u_{j}^{*}u_{k} , α j k ∈ C \alpha _{jk} \in \mathbb {C} . We prove that if f f is self-adjoint and T r ( f ( U 1 , U 2 , U 3 ) ) ≥ 0 \mathrm {Tr}(f(U_{1}, U_2 ,U_{3}))\ge 0 for arbitrary unitary matrices U 1 , U 2 , U 3 U_{1}, U_2, U_3 , then f f is a sum of hermitian squares. To prove this statement we reduce it to the question whether a certain Tarski sentence is true. Tarski’s decidability theorem thus provides an algorithm to answer this question. We use an algorithm due to Lazard and Rouillier for computing the number of real roots of a parametric system of polynomial equations and inequalities implemented in Maple to check that the Tarski sentence is true. As an application, we describe the set of parameters a 1 , a 2 , a 3 , a 4 a_1, a_2, a_3, a_4 such that there are unitary operators U 1 , … , U 4 U_1, \ldots , U_4 connected by the linear relation a 1 U 1 + a 2 U 2 + a 3 U 3 + a 4 U 4 = 0 a_1 U_1+a_2 U_2 +a_3 U_3 +a_4 U_4 =0 .

In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic number field or a totally definite rational quaternion algebra. This inequality implies that the 16-dimensional Barnes-Wall lattice has optimal density among all 16-dimensional lattices with Hurwitz structures.

K. Ball has proved the “complex plank problem”: if ( x k ) k = 1 n \left (x_{k}\right )_{k=1}^{n} is a sequence of norm 1 1 vectors in a complex Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) \left (H, \, \langle \cdot ,\cdot \rangle \right ) , then there exists a unit vector x x for which \[ | ⟨ x , x k ⟩ | ≥ 1 / n , k = 1 , … , n . \left |\langle {x}, x_{k}\rangle \right |\geq 1/\sqrt {n}\,,\quad k=1, \ldots , n\,. \] In general, this result is not true on real Hilbert spaces. However, in special cases we prove that the same result holds true. In general, for some unit vector x x we have derived the estimate \[ | ⟨ x , x k ⟩ | ≥ max { λ 1 / n , 1 / λ n n } , \left |\langle {x}, x_{k}\rangle \right | \geq \max \left \{\sqrt {\lambda _{1}/n},\, 1/\sqrt {\lambda _{n}n}\right \}\,, \] where λ 1 \lambda _{1} is the smallest and λ n \lambda _{n} is the largest eigenvalue of the Hermitian matrix A = [ ⟨ x j , x k ⟩ ] A=\left [\langle {x_{j}}, x_{k}\rangle \right ] , j , k = 1 , … , n j, k=1, \ldots , n . We have also improved known estimates for the norms of homogeneous polynomials which are products of linear forms on real Hilbert spaces.

In an earlier paper we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, including all connected symmetric pairs over C. In this paper we show that the pairs (GL(V),O(V)), (GL(V),U(V)), (U(V), O(V)), (O(V ⊕ W), O(V) x O(W)), (U(V ⊕ W), U(V) × U(W)) are regular, where V and W are quadratic or Hermitian spaces over an arbitrary local field of characteristic zero. We deduce from this that the pairs (GL n (ℂ), O n (ℂ)) and (O n+m (ℂ),O n (ℂ) × O m (ℂ)) are Gelfand pairs.

We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $\mathbb{Q}(\sqrt{-m})$ for all m. For each imaginary quadratic field $\mathbb{Q}(\sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13,14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneeberger's fifteen theorem and ours is the number 13., Comment: 20 Pages

It is shown that over infinitely many imaginary quadratic fields there exists a Hermitian lattice in all even ranks n ≥ 2 n \ge 2 which is generated by its 4 n 4n minimal vectors but which is not generated by 2 n − 1 2n-1 minimal vectors.

In the present note we describe a family of H ∗ H^* -algebra structures on the set L 2 ( X ) L^2(X) of square integrable functions on a rank-one para-Hermitian symmetric space X X .

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kaehler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived., Comment: 12 pages, AMSLatex

Consider the ring of all κ × κ \kappa \times \kappa column finite matrices over a ring R R . We prove that each such matrix is conjugate to a row and column finite matrix if and only if R R is right Noetherian and κ \kappa is countable. We then demonstrate that one can perform this conjugation on countably many matrices simultaneously. Some applications and limitations are given.

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

The gamma, beta and Dirichlet functions have been generalized in several ways by Ingham, Siegel, Bellman and Olkin. These authors defined them as integrals having the integrand as a scalar function of real symmetric matrix. In this article, we have defined and studied these functions when the integrand is a scalar function of Hermitian matrix.

In this paper we describe the general forms of all (nonlinear) continuous functionals on the sets of positive definite, positive semi-definite and Hermitian matrices which are multiplicative on the commuting elements. As a consequence, we obtain some new characterizations of the determinant on those classes of matrices.

In this paper we generalize the notion of p p -adic modular form to the Hermitian modular case and prove a formula that shows a coincidence between certain p p -adic Hermitian Eisenstein series and the genus theta series associated with Hermitian matrix with determinant p p .

We present and analyze homotopic (continuation) residual correction algorithms for the computation of matrix inverses. For complex indefinite Hermitian input matrices, our homotopic methods substantially accelerate the known nonhomotopic algorithms. Unlike the nonhomotopic case our algorithms require no pre-estimation of the smallest singular value of an input matrix. Furthermore, we guarantee rapid convergence to the inverses of well-conditioned structured matrices even where no good initial approximation is available. In particular we yield the inverse of a well-conditioned n × n n \times n matrix with a structure of Toeplitz/Hankel type in O ( n log 3 ⁡ n ) O(n\log ^3n) flops. For a large class of input matrices, our methods can be extended to computing numerically the generalized inverses. Our numerical experiments confirm the validity of our analysis and the efficiency of the presented algorithms for well-conditioned input matrices and furnished us with the proper values of the parameters that define our algorithms.

Let G G be a finite group generated by unitary reflections in a Hermitian space V V , and let ζ \zeta be a root of unity. Let E E be a subspace of V V , maximal with respect to the property of being a ζ \zeta -eigenspace of an element of G G , and let C C be the parabolic subgroup of elements fixing E E pointwise. If χ \chi is any linear character of G G , we give a condition for the restriction of χ \chi to C C to be trivial in terms of the invariant theory of G G , and give a formula for the polynomial ∑ x ∈ G χ ( x ) T d ( x , ζ ) \sum _{x\in G}\chi (x)T^{d(x,\zeta )} , where d ( x , ζ ) d(x,\zeta ) is the dimension of the ζ \zeta -eigenspace of x x . Applications include criteria for regularity, and new connections between the invariant theory and the structure of G G .

We consider the quadratic eigenvalue problem (or the QEP) (λ 2 A + λB + C)x = 0, where A, B, and C are Hermitian with A positive definite. The QEP is called hyperbolic if (x*Bx) 2 > 4(x*Ax)(x*Cx) for all nonzero x ∈ C R . We show that a relatively efficient test for hyperbolicity can be obtained by computing the eigenvalues of the QEP. A hyperbolic QEP is overdamped if B is positive definite and C is positive semidefinite. We show that a hyperbolic QEP (whose eigenvalues are necessarily real) is overdamped if and only if its largest eigenvalue is nonpositive. For overdamped QEPs, we show that all eigenpairs can be found efficiently by finding two solutions of the corresponding quadratic matrix equation using a method based on cyclic reduction. We also present a new measure for the degree of hyperbolicity of a hyperbolic QEP.

We find full invariants for detecting non--zero elements in ${L}_0(F(t))\otimes \Q$, this group plays an important role in topology in the work done by Casson and Gordon., Comment: 8 pages