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} .     

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 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 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.     

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.     

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.     

Multipliers of weighted semigroups and associated Beurling Banach algebras

Given a weighted discrete abelian semigroup (S, ω), the semigroup M _ω (S) of ω-bounded multipliers as well as the Rees quotient M _ω (S)/S together with their respective weights [(w)\tilde]\tilde{\omega} and [(w)\tilde]_q\tilde{\omega}_q induced by ω are studied; for a large class of weights ω, the quotient l¹(M_w(S),[(w)\tilde])/l¹(S,w)\ell^1(M_{\omega}(S),\tilde{\omega})/\ell^1(S,{\omega}) is realized as a Beurling algebra on the quotient semigroup M _ω (S)/S; the Gel’fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

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.     

Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system

Join-Irreducible Boolean Functions

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset [(W)\tilde]\tilde{\Omega}. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield join-irreducible members of [(W)\tilde]\tilde{\Omega} are the − 2-monomorphic Steiner systems. We also describe the graphs which correspond to join-irreducible members of [(W)\tilde]\tilde{\Omega}.     

A polynomial path-following interior point algorithm for general linear complementarity problems

Linear Complementarity Problems (LCPs) belong to the class of \mathbbNP{\mathbb{NP}} -complete problems. Therefore we cannot expect a polynomial time solution method for LCPs without requiring some special property of the coefficient matrix. Our aim is to construct interior point algorithms which, according to the duality theorem in EP (Existentially Polynomial-time) form, in polynomial time either give a solution of the original problem or detects the lack of property P_*([(k)\tilde]){\mathcal{P}_*(\tilde\kappa)} , with arbitrary large, but apriori fixed [(k)\tilde]{\tilde\kappa}). In the latter case, the algorithms give a polynomial size certificate depending on parameter [(k)\tilde]{\tilde{\kappa}} , the initial interior point and the input size of the LCP). We give the general idea of an EP-modification of interior point algorithms and adapt this modification to long-step path-following interior point algorithms.     

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}}.     

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Schlicht envelopes of holomorphy and topology

Let Ω be a domain in ${\mathbb{C}^{2}}Let Ω be a domain in \mathbbC²{\mathbb{C}^{2}}, and let p: [(W)\tilde]? \mathbbC²{\pi: \tilde{\Omega}\rightarrow \mathbb{C}^{2}} be its envelope of holomorphy. Also let W￠=p([(W)\tilde]){\Omega'=\pi(\tilde{\Omega})} with i: W\hookrightarrow W￠{i: \Omega \hookrightarrow \Omega'} the inclusion. We prove the following: if the induced map on fundamental groups i_*:p₁(W) ? p₁(W￠){i_{*}:\pi_{1}(\Omega) \rightarrow \pi_{1}(\Omega')} is a surjection, and if π is a covering map, then Ω has a schlicht envelope of holomorphy. We then relate this to earlier work of Fornaess and Zame.     

Envelope of holomorphy of a tube domain over a complex Lie group

According to S. Bochner [6, 7]: IfD =B +iℝ ⁿ is a tube domain in ℂ ⁿ , where B is a domain in ℝ ⁿ , and if [(B)\tilde]\tilde B is the convex envelope of B, then any holomorphic function on D extends to the tube domain [(D)\tilde] = [(B)\tilde] + i\mathbbRⁿ \tilde D = \tilde B + i\mathbb{R}^n , which is a univalent envelope of holomorphy of D. We give a generalization of this result to (nonunivalent) tube domains over a complex Lie group which admit a closed sub-group as a real form. Application: If (V, φ) is a tube domain over ℂ ⁿ and if B is the convex envelope of ϕ(V)∩ℝ ⁿ in ℝ ⁿ , then [(V)\tilde] = B + i\mathbbRⁿ \tilde V = B + i\mathbb{R}^n is an envelope of holomorphy of (V, φ).     

Maximizing Strictly Convex Quadratic Functions with Bounded Perturbations

The problem of maximizing [(f)\tilde]=f+p\tilde{f}=f+p over some convex subset D of the n-dimensional Euclidean space is investigated, where f is a strictly convex quadratic function and p is assumed to be bounded by some s∈[0,+∞[. The location of global maximal solutions of [(f)\tilde]\tilde{f} on D is derived from the roughly generalized convexity of [(f)\tilde]\tilde{f}. The distance between global (or local) maximal solutions of [(f)\tilde]\tilde{f} on D and global (or local, respectively) maximal solutions of f on D is estimated. As consequence, the set of global (or local) maximal solutions of [(f)\tilde]\tilde{f} on D is upper (or lower, respectively) semicontinuous when the upper bound s tends to zero.     

A New Recursion in the Theory of Macdonald Polynomials

The bigraded Frobenius characteristic of the Garsia-Haiman module M _μ is known [7, 10] to be given by the modified Macdonald polynomial [(H)\tilde]_m[X; q, t]{\tilde{H}_{\mu}[X; q, t]}. It follows from this that, for m\vdash n{\mu \vdash n} the symmetric polynomial ?_p1 [(H)\tilde]_m[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t]} is the bigraded Frobenius characteristic of the restriction of M _μ from S _n to S _n-1. The theory of Macdonald polynomials gives explicit formulas for the coefficients c _{μ v} occurring in the expansion ?_p1 [(H)\tilde]_m[X; q, t] = ?_{v ? m}c_mv [(H)\tilde]_v[X; q, t]{{\partial_{p1}} \tilde{H}_{\mu}[X; q, t] = \sum_{v \to \mu}c_{\mu v} \tilde{H}_{v}[X; q, t]}. In particular, it follows from this formula that the bigraded Hilbert series F μ (q, t) of M _μ may be calculated from the recursion F_m (q, t) = ?_{v ? m}c_mv F_v (q, t){F_\mu (q, t) = \sum_{v \to \mu}c_{\mu v} F_v (q, t)}. One of the frustrating problems of the theory of Macdonald polynomials has been to derive from this recursion that Fm(q, t) ? N[q, t]{F\mu (q, t) \in \mathbf{N}[q, t]}. This difficulty arises from the fact that the c _{μ v} have rather intricate expressions as rational functions in q, t. We give here a new recursion, from which a new combinatorial formula for F _μ (q, t) can be derived when μ is a two-column partition. The proof suggests a method for deriving an analogous formula in the general case. The method was successfully carried out for the hook case by Yoo in [15].     

A criterion for weak convergence on Berkovich projective space

We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space P^N_K{{\bf P}^{N}_K} over a complete non-archimedean field K. As an application, we give a sufficient condition for a certain type of equidistribution on P^N_K{{\bf P}^{N}_K} in terms of a weak Zariski-density property on the scheme-theoretic projective space \mathbb P^N_[(K)\tilde]{{\mathbb P}^N_{\tilde{K{}_{\vphantom{0}}}}} over the residue field [(K)\tilde]{\tilde{K}} . As a second application, in the case of residue characteristic zero we give an ergodic-theoretic equidistribution result for the powers of a point a in the N-dimensional unit torus \mathbb T^N_K{{\mathbb T}^N_K} over K. This is a non-archimedean analogue of a well-known result of Weyl over \mathbb C{\mathbb C} , and its proof makes essential use of a theorem of Mordell-Lang type for \mathbb G_m^N{{\mathbb G}_m^N} due to Laurent.