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)}.     

Minimal rotational hypersurfaces in Minkowski (<Emphasis Type="Italic">α</Emphasis>, <Emphasis Type="Italic">β</Emphasis>)-space

In Finsler geometry, minimal surfaces with respect to the Busemann-Hausdorff measure and the Holmes-Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. In this paper, we give the explicit expressions of BH-minimal and HT-minimal rotational hypersurfaces generated by plane curves rotating around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski (α, β)-space (\mathbbVⁿ⁺¹,[(F_b)\tilde]){(\mathbb{V}^{n+1},\tilde{F_b})} , where \mathbbVⁿ⁺¹{\mathbb{V}^{n+1}} is an (n+1)-dimensional real vector space, [(F_b)\tilde]=[(a)\tilde]f([(b)\tilde]/[(a)\tilde]), [(a)\tilde]{\tilde{F_b}=\tilde{\alpha}\phi(\tilde{\beta}/\tilde{\alpha}), \tilde{\alpha}} is the Euclidean metric, [(b)\tilde]{\tilde{\beta}} is a one form of constant length b:=||[(b)\tilde]||_[(a)\tilde], [(b)\tilde]^\sharp{b:=\|\tilde{\beta}\|_{\tilde{\alpha}}, \tilde{\beta}^{\sharp}} is the dual vector of [(b)\tilde]{\tilde{\beta}} with respect to [(a)\tilde]{\tilde{\alpha}} . As an application, we first give the explicit expressions of the forward complete BH-minimal rotational surfaces generated around the axis in the direction of [(b)\tilde]^\sharp{\tilde{\beta}^{\sharp}} in Minkowski Randers 3-space (\mathbbV³,[(a)\tilde]+[(b)\tilde]){(\mathbb{V}^{3},\tilde{\alpha}+\tilde{\beta})} .     

On quasiorder lattices and topology lattices of algebras

In this paper, it is shown that the dual [(\textQord)\tilde]\mathfrakA \widetilde{\text{Qord}}\mathfrak{A} of the quasiorder lattice of any algebra \mathfrakA \mathfrak{A} is isomorphic to a sublattice of the topology lattice á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . Further, if \mathfrakA \mathfrak{A} is a finite algebra, then [(\textQord)\tilde]\mathfrakA @ á( \mathfrakA ) \widetilde{\text{Qord}}\mathfrak{A} \cong \Im \left( \mathfrak{A} \right) . We give a sufficient condition for the lattices [(\textCon)\tilde]\mathfrakA\text, [(\textQord)\tilde]\mathfrakA \widetilde{\text{Con}}\mathfrak{A}{\text{,}} \widetilde{\text{Qord}}\mathfrak{A} , and á( \mathfrakA ) \Im \left( \mathfrak{A} \right) . to be pairwise isomorphic. These results are applied to investigate topology lattices and quasiorder lattices of unary algebras.     

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}}.     

On some Dirichlet series related to the Riemann zeta function,I

On the assumption of the truth of the Riemann hypothesis for the Riemann zeta function we construct a class of modified von-Mangoldt functions with slightly better mean value properties than the well known function L\Lambda . For every e ? (0,1/2)\varepsilon \in (0,1/2) there is a [(L)\tilde] : \Bbb N ? \Bbb C\tilde {\Lambda} : \Bbb N \to \Bbb C such that¶ i) [(L)\tilde] (n) = L (n) (1 + O(n^-1/4  logn))\tilde {\Lambda} (n) = \Lambda (n) (1 + O(n^{-1/4\,} \log n)) and¶ii) ?_{n \leqq x} [(L)\tilde] (n) (1- [(n)/(x)]) = [(x)/2] + O(x^1/4+e) (x \geqq 2).\sum \limits_{n \leqq x} \tilde {\Lambda} (n) \left(1- {{n}\over{x}}\right) = {{x}\over{2}} + O(x^{1/4+\varepsilon }) (x \geqq 2).¶Unfortunately, this does not lead to an improved error term estimation for the unweighted sum ?_{n \leqq x} [(L)\tilde] (n)\sum \limits_{n \leqq x} \tilde {\Lambda} (n), which would be of importance for the distance between consecutive primes.     

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let (M,[(g)\tilde]){(\mathcal {M},\tilde{g})} be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation

The Dual Quiver of the Auslander–Reiten Quiver of Path Algebras

Let Q be a finite quiver of type A _n , n ≥ 1, D _n , n ≥ 4, E ₆, E ₇ and E ₈, σ ∈ Aut(Q), k be an algebraic closed field whose characteristic does not divide the order of σ. In this article, we prove that the dual quiver [(G_Q)\tilde]\widetilde{\Gamma_{Q}} of the Auslander–Reiten quiver Γ _Q of kQ, the Auslander–Reiten quiver of kQ#kás?kQ\#k\langle\sigma\rangle, and the Auslander–Reiten quiver G_[(Q)\tilde]\Gamma_{\widetilde{Q}} of k[(Q)\tilde]k\widetilde{Q}, where [(Q)\tilde]\widetilde{Q} is the dual quiver of Q, are isomorphic.     

Study of Some Fundamental Projective Quadrics Associated with a Standard Pseudo-Hermitian Space H p,q

This self-contained short note deals with the study of the properties of some real projective compact quadrics associated with a a standard pseudo-hermitian space H _p,q , namely [(Q(p, q))\tilde], [(Q_2p+1,1)\tilde], [(Q_1,2q+1)\tilde], [(H_p,q)\tilde].  [(Q(p, q))\tilde]{\widetilde{Q(p, q)}, \widetilde{Q_{2p+1,1}}, \widetilde{Q_{1,2q+1}}, \widetilde{H_{p,q}}. \, \widetilde{Q(p, q)}} is the (2n – 2) real projective quadric diffeomorphic to (S ^2p–1 × S ^2q–1)/Z ₂. inside the real projective space P(E ₁), where E ₁ is the real 2n-dimensional space subordinate to H _p,q . The properties of [(Q(p, q))\tilde]{\widetilde{Q(p, q)}} are investigated. [(H_p,q)\tilde]{\widetilde{H_p,q}} is the real (2n – 3)-dimensional compact manifold-(projective quadric)- associated with H _p,q , inside the complex projective space P(H _p,q ), diffeomorphic to (S ^2p–1 × S ^2q–1)/S ¹. The properties of [(H_p,q)\tilde]{\widetilde{H_{p,q}}} are studied. [(Q_2p+1,1)\tilde]{\widetilde{Q_{2p+1,1}}} is a 2p-dimensional standard real projective quadric, and [(Q_1,2q+1)\tilde]{\widetilde{Q_{1,2q+1}}} is another standard 2q-dimensional projective quadric. [(Q_2p+1,1)\tilde] è[(Q_1,2q+1)\tilde]{\widetilde{Q_{2p+1,1}} \cup \widetilde{Q_{1,2q+1}}}, union of two compact quadrics plays a part in the understanding of the "special pseudo-unitary conformal compactification" of H _p,q . It is shown how a distribution y → D _y , where y ? H\{0},H{y \in H\backslash\{0\},H} being the isotropic cone of H _p,q allows to [(H_p+1,q+1)\tilde]{\widetilde{H_{p+1,q+1}}} to be considered as a "special pseudo-unitary conformal compactified" of H _p,q × R. The following results precise the presentation given in [1,c].     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

On Li’s coefficients for the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-functions

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π ^′ of GL_m(\mathbbA_F)GL_{m}(\mathbb{A}_{F}) and GL_m^￠(\mathbbA_F)GL_{m^{\prime }}(\mathbb{A}_{F}) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) attached to L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) . Then, we deduce a full asymptotic expansion of the archimedean contribution to l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) , the nth Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μ _π (v,j) of π. Namely, we prove that under GRH for L(s,p_f×[(p)\tilde]_f)L(s,\pi _{f}\times \widetilde{\pi}_{f}) one has |Rem_p(v,j)| ￡ \frac14|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4} for all archimedean places v at which π is unramified and all j=1,…,m.     

A new class of integrable systems

This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=¹/₂(x₁²+x₂²) +V(y₁,y₂) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y₁,y₂)=¹/₂(a₁y₁²+a₂y₂²) + ¹/₄b₁y₁⁴ + ¹/₄b₂y₂⁴ + ¹/₂b₃y₁²y₂² + ?_k=1³g_k(y₁²+y₂²) ^k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a₁,a₂,b₁,b₂,b₃,g₁,g₂,g₃) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .     

On Ricci curvature of isotropic and Lagrangian submanifolds in complex space forms

Let M ⁿ be a Riemannian n-manifold. Denote by S(p) and [`(Ric)](p)\overline {Ric}(p) the Ricci tensor and the maximum Ricci curvature on M <sup>n</sup> at a point p ? M<sup>n</sup>p\in M^n, respectively. First we show that every isotropic submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies S ￡ ((n-1)c+ [(n<sup>2</sup>)/4] H<sup>2</sup>)gS\leq ((n-1)c+ {n^2 \over 4} H^2)g, where H<sup>2</sup> and g are the squared mean curvature function and the metric tensor on M <sup>n</sup>, respectively. The equality case of the above inequality holds identically if and only if either M <sup>n</sup> is totally geodesic submanifold or n = 2 and M <sup>n</sup> is a totally umbilical submanifold. Then we prove that if a Lagrangian submanifold of a complex space form [(M)\tilde]<sup>m</sup>(4 c)\widetilde M^m(4\,c) satisfies [`(Ric)] = (n-1)c+ [(n²)/4] H²\overline {Ric}= (n-1)c+ {n^2 \over 4} H^2 identically, then it is a minimal submanifold. Finally, we describe the geometry of Lagrangian submanifolds which satisfy the equality under the condition that the dimension of the kernel of second fundamental form is constant.     

A note on splitting-type variational problems with subquadratic growth

We consider variational problems of splitting-type, i.e., we want to minimize

Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

Let W ì \mathbb C^d{\Omega \subset{\mathbb C}^{d}} be an irreducible bounded symmetric domain of type (r, a, b) in its Harish–Chandra realization. We study Toeplitz operators Tⁿ_g{T^{\nu}_{g}} with symbol g acting on the standard weighted Bergman space H_n²{H_\nu^2} over Ω with weight ν. Under some conditions on the weights ν and ν ₀ we show that there exists C(ν, ν ₀) > 0, such that the Berezin transform [(g)\tilde]_n0{\tilde{g}_{\nu_{0}}} of g with respect to H²_n0{H^2_{\nu_0}} satisfies:

On the 3-Torsion Part of the Homology of the Chessboard Complex

Let 1 ≤ m ≤ n. We prove various results about the chessboard complex M _m,n , which is the simplicial complex of matchings in the complete bipartite graph K _m,n . First, we demonstrate that there is nonvanishing 3-torsion in [(H)\tilde]_d(\sf M_m,n; \mathbb Z){{\tilde{H}_d({\sf M}_{m,n}; {\mathbb Z})}} whenever \fracm+n-43 ￡ d ￡ m-4{{\frac{m+n-4}{3}\leq d \leq m-4}} and whenever 6 ≤ m < n and d = m − 3. Combining this result with theorems due to Friedman and Hanlon and to Shareshian and Wachs, we characterize all triples (m, n, d ) satisfying [(H)\tilde]_d (\sf M_m,n; \mathbb Z) 1 0{{\tilde{H}_d \left({\sf M}_{m,n}; {\mathbb Z}\right) \neq 0}}. Second, for each k ≥ 0, we show that there is a polynomial f _k (a, b) of degree 3k such that the dimension of [(H)\tilde]_k+a+2b-2 (\sf M_{k+a+3b-1,k+2a+3b-1}; \mathbb Z₃){{\tilde{H}_{k+a+2b-2}}\,\left({{\sf M}_{k+a+3b-1,k+2a+3b-1}}; \mathbb Z_{3}\right)}, viewed as a vector space over \mathbbZ₃{\mathbb{Z}_3}, is at most f _k (a, b) for all a ≥ 0 and b ≥ k + 2. Third, we give a computer-free proof that [(H)\tilde]₂ (\sf M_5,5; \mathbb Z) @ \mathbb Z₃{{\tilde{H}_2 ({\sf M}_{5,5}; \mathbb {Z})\cong \mathbb Z_{3}}}. Several proofs are based on a new long exact sequence relating the homology of a certain subcomplex of M _m,n to the homology of M _m-2,n-1 and M _m-2,n-3.     

Tight Lower Bounds for Halfspace Range Searching

We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in ℝ ^d , where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space requirements, we prove a lower bound for general semigroups of [\varOmega\tilde](n^1-1/(d+1)/m^1/(d+1))\widetilde{\varOmega}(n^{1-1/(d+1)}/m^{1/(d+1)}) and for integral semigroups of [\varOmega\tilde](n/m^1/d)\widetilde{\varOmega}(n/m^{1/d}).     

Maps Between Uniform Algebras Preserving Norms of Rational Functions

Let A, B be uniform algebras. Suppose that A ₀, B ₀ are subgroups of A ⁻¹, B ⁻¹ that contain exp A, exp B respectively. Let α be a non-zero complex number. Suppose that m, n are non-zero integers and d is the greatest common divisor of m and n. If T : A ₀ → B ₀ is a surjection with ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f,g ? A₀{f,g \in A_0}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? A₀{f \in A_0}. This result leads to the following assertion: Suppose that S _A , S _B are subsets of A, B that contain A ⁻¹, B ⁻¹ respectively. If m, n > 0 and a surjection T : S _A → S _B satisfies ||T(f)^mT(g)ⁿ - a||_￥ = ||f^mgⁿ - a||_￥{\|T(f)^{m}T(g)^{n} - \alpha\|_{\infty} = \|f^{m}g^{n} - \alpha\|_{\infty}} for all f, g ? S_A{f, g \in S_A}, then there exists a real-algebra isomorphism [(T)\tilde] : A ? B{\tilde{T} : A \rightarrow B} such that [(T)\tilde](f)^d = (T(f)/T(1))^d{\tilde{T}(f)^d = (T(f)/T(1))^d} for every f ? S_A{f \in S_A}. Note that in these results and elsewhere in this paper we do not assume that T(exp A) = exp B.     

Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

A Note on Aluthge Transforms of Complex Symmetric Operators and Applications

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator [(T)\tilde] = |T|^\frac12 U|T|^\frac12\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}} is complex symmetric, (ii) if T is a complex symmetric operator, then ([(T)\tilde])^*(\tilde{T})^{*} and [(T^*)\tilde]\widetilde{T^{*}} are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then [((T^*))\tilde]_s,t\widetilde{(T^{*})}_{s,t} and ([(T)\tilde]_t,s)^*(\tilde{T}_{t,s})^{*} are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H _{\mathfrakg\mathfrak{g} ) of the tangent Lie algebra \mathfrakg\mathfrak{g} of the group G with coefficients in the one-dimensional representation \mathfrakg\mathfrak{g} \mathbbK\mathbb{K} defined by [(W)\tilde] \mathfrakg \tilde \Omega _\mathfrak{g} of H ¹(G/ \mathfrakg\mathfrak{g} .}