The Idempotents of the TL n -Module {\otimes^{n}\mathbb{C}2} in Terms of Elements of {{\rm U}_{q} \mathfrak{sl}_2}

The vector space \({\otimes^{n}\mathbb{C}^2}\) upon which the XXZ Hamiltonian with n spins acts bears the structure of a module over both the Temperley–Lieb algebra \({{\rm TL}_{n}(\beta = q + q^{-1})}\) and the quantum algebra \({{\rm U}_{q} \mathfrak{sl}_2}\) . The decomposition of \({\otimes^{n}\mathbb{C}^2}\) as a \({{\rm U}_{q} \mathfrak{sl}_2}\) -module was first described by Rosso (Commun Math Phys 117:581–593, 1988), Lusztig (Cont Math 82:58–77, 1989) and Pasquier and Saleur (Nucl Phys B 330:523–556, 1990) and that as a TL _n -module by Martin (Int J Mod Phys A 7:645–673, 1992) (see also Read and Saleur Nucl Phys B 777(3):316–351, 2007; Gainutdinov and Vasseur Nucl Phys B 868:223–270, 2013). For q generic, i.e. not a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) is known to be a sum of irreducible modules. We construct the projectors (idempotents of the algebra of endomorphisms of \({\otimes^{n}\mathbb{C}^2}\) ) onto each of these irreducible modules as linear combinations of elements of \({{\rm U}_{q} \mathfrak{sl}_2}\) . When q = q _c is a root of unity, the TL _n -module \({\otimes^{n}\mathbb{C}^2}\) (with n large enough) can be written as a direct sum of indecomposable modules that are not all irreducible. We also give the idempotents projecting onto these indecomposable modules. Their expression now involves some new generators, whose action on \({\otimes^{n}\mathbb{C}^2}\) is that of the divided powers \({(S^{\pm})^{(r)} = \lim_{q \rightarrow q_{c}} (S^{\pm})^r/[r]!}\) .     

Changing-sign bubble solutions for an anisotropic sinh-Poisson equation

We consider the following anisotropic sinh-Poisson equation $${\rm div} (a(x) \nabla u)+ 2\varepsilon^2 a(x) {\rm sinh}\,u=0\ \ {\rm in}\ \Omega, \quad u=0 \ \ {\rm on}\ \partial \Omega,$$ where ${\Omega \subset \mathbb{R}^2}$ is a bounded smooth domain and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient ${a(x)}$ on the existence of bubbling solutions. We show that there exists a family of solutions u _?? concentrating positively and negatively at ${\bar{x}}$ , a given local critical point of a(x), for ?? sufficiently small, for which with the property $$2\varepsilon^2a(x){\rm sinh} u_\varepsilon \rightharpoonup 8\pi\sum\limits_{j=1}^{m}b_j\delta_{\bar{x}},$$ where ${b_j=\pm 1}$ . This result shows a striking difference with the isotropic case (a(x) ?? Constant) in Bartolucci and Pistoia (IMA J Appl Math 72(6):706?C729, 2007), Jost et?al. (Calc Var Partial Differ Equ 31:263?C276, 2008) and Esposito and Wei (Calc Var Partial Differ Equ 34:341?C375, 2009).     

Circular Coloring of Planar Digraphs

Let D be a digraph. The circular chromatic number ${\chi_c(D)}$ and chromatic number ${\chi(D)}$ of D were proposed recently by Bokal et?al. Let ${\vec{\chi_c}(G)={\rm max}\{\chi_c(D)| D\, {\rm is\, an\, orientation\, of} G\}}$ . Let G be a planar graph and n?≥ 2. We prove that if the girth of G is at least ${\frac{10n-5}{3},}$ then ${\vec{\chi_c}(G)\leq \frac{n}{n-1}}$ . We also study the circular chromatic number of some special planar digraphs.     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$ where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\) . Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.     

New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane

In the projective planes PG(2, q), more than 1230 new small complete arcs are obtained for ${q \leq 13627}$ and ${q \in G}$ where G is a set of 38 values in the range 13687,..., 45893; also, ${2^{18} \in G}$ . This implies new upper bounds on the smallest size t ₂(2, q) of a complete arc in PG(2, q). From the new bounds it follows that $$t_{2}(2, q) < 4.5\sqrt{q} \, {\rm for} \, q \leq 2647$$ and q = 2659,2663,2683,2693,2753,2801. Also, $$t_{2}(2, q) < 4.8\sqrt{q} \, {\rm for} \, q \leq 5419$$ and q = 5441,5443,5449,5471,5477,5479,5483,5501,5521. Moreover, $$t_{2}(2, q) < 5\sqrt{q} \, {\rm for} \, q \leq 9497$$ and q = 9539,9587,9613,9623,9649,9689,9923,9973. Finally, $$t_{2}(2, q) <5 .15\sqrt{q} \, {\rm for} \, q \leq 13627$$ and q = 13687,13697,13711,14009. Using the new arcs it is shown that $$t_{2}(2, q) < \sqrt{q}\ln^{0.73}q {\rm for} 109 \leq q \leq 13627\, {\rm and}\, q \in G.$$ Also, as q grows, the positive difference ${\sqrt{q}\ln^{0.73} q-\overline{t}_{2}(2, q)}$ has a tendency to increase whereas the ratio ${\overline{t}_{2}(2, q)/(\sqrt{q}\ln^{0.73} q)}$ tends to decrease. Here ${\overline{t}_{2}(2, q)}$ is the smallest known size of a complete arc in PG(2,q). These properties allow us to conjecture that the estimate ${t_{2}(2,q) < \sqrt{q}\ln ^{0.73}q}$ holds for all ${q \geq 109.}$ The new upper bounds are obtained by finding new small complete arcs in PG(2,q) with the help of a computer search using randomized greedy algorithms. Finally, new forms of the upper bound on t ₂(2,q) are proposed.     

Behaviour at infinity of solutions of some linear functional equations in normed spaces

Let \({\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}, I = (d, \infty), \phi : I \to I}\) be unbounded continuous and increasing, X be a normed space over \({\mathbb{K}, \mathcal{F} : = \{f \in X^I : {\rm lim}_{t \to \infty} f(t) {\rm exists} \, {\rm in} X\},\hat{a} \in \mathbb{K}, \mathcal{A}(\hat{a}) : = \{\alpha \in \mathbb{K}^I : {\rm lim}_{t \to \infty} \alpha(t) = \hat{a}\},}\) and \({\mathcal{X} : = \{x \in X^I : {\rm lim} \, {\rm sup}_{t \to \infty} \|x(t)\| < \infty\}}\) . We prove that the limit lim _{t → ∞} x(t) exists for every \({f \in \mathcal{F}, \alpha \in \mathcal{A}(\hat{a})}\) and every solution \({x \in \mathcal{X}}\) of the functional equation $$x(\phi(t)) = \alpha(t) x(t) + f(t)$$ if and only if \({|\hat{a}| \neq 1}\) . Using this result we study behaviour of bounded at infinity solutions of the functional equation $$x(\phi^{[k]}(t)) = \sum_{j=0}^{k-1} \alpha_j(t) x (\phi^{[j]}(t)) + f(t),$$ under some conditions posed on functions \({\alpha_j(t), j = 0, 1,\ldots, k - 1,\phi}\) and f.     

On the generalization of Faltings’ Annihilator Theorem

Let R be a commutative Noetherian ring, and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the invariants ${\inf\{i \in \mathbb{N}_0|\, \rm{dim\, Supp}(\mathfrak{b}^t H_{\mathfrak{a}}^i(M)) \geq n\, \rm{for\, all}\, t \in \mathbb{N}_0\}}$ and ${\inf\{\lambda_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\, \mathfrak{p} \in {\rm Spec} \, R\, \rm{and\, dim}\, R/ \mathfrak{p} \geq n\}}$ are equal, for every finitely generated R-module M and for all ideals ${\mathfrak{a}, \mathfrak{b}}$ of R with ${\mathfrak{b}\subseteq \mathfrak{a}}$ . This generalizes Faltings’ Annihilator Theorem (see [6]).     

Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

On Transversally Elliptic Operators and the Quantization of Manifolds with f-Structure

Given an f-structure ${\varphi}$ on a manifold M, together with a compatible metric g and connection ${\nabla}$ on M, we construct an odd firstorder differential operator D whose principal symbol is of the type considered in [13]. In the special case of a CR-integrable almost ${\mathcal {S}}$ -structure, we show that when ${\nabla}$ is the generalized Tanaka-Webster connection of Lotta and Pastore, the operator D is given by D = ${{\sqrt {2} (\overline {\partial}_b + \overline{\partial}_{b}^{\ast})}}$ , where ${\overline {\partial}_b}$ is the tangential Cauchy-Riemann operator. We then describe two types of “quantization” of manifolds with f-structure that reduce to familiar methods in symplectic geometry in the case that ${\varphi}$ is a compatible almost complex structure, and to the contact quantizations defined in [16] when ${\varphi}$ comes from a contact metric structure.     

Geometric Sufficient Conditions for Compactness of the Complex Green Operator

We establish compactness estimates for $\overline{\partial}_{M}$ on a compact pseudoconvex CR-submanifold M of ? ⁿ of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the $\overline{\partial}$ -Neumann operator given in (Straube in Ann. Inst. Fourier, 54(3):699?C710, 2004; Munasinghe and Straube in Pac. J. Math., 232(2):343?C354 2007). These conditions are formulated in terms of certain short time flows in complex tangential directions.     

Differentiable vectors and unitary representations of Fréchet–Lie supergroups

A locally convex Lie group G has the Trotter property if, for every $x_1, x_2 \in \mathfrak{g }$ , $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$ holds uniformly on compact subsets of $\mathbb{R }$ . All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if G has the Trotter property, $\pi : G \rightarrow {\mathrm{GL}}(V)$ is a continuous representation of G on a locally convex space, and $v \in V$ is a vector such that $\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v$ exists for every $x \in \mathfrak{g }$ , then the map $\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v$ is linear. Using this result we conclude that, for a representation of a locally exponential Fréchet–Lie group G on a metrizable locally convex space, the space of $\mathcal{C }^{k}$ -vectors coincides with the common domain of the k-fold products of the operators $\overline{\mathtt{d}\pi }(x)$ . For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Fréchet–Lie supergroups $(G,\mathfrak{g })$ where G has the Trotter property, the common domain of the operators of $\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}$ can always be extended to the space of smooth (resp., analytic) vectors for G.     

A Reduction Theorem for the Kripke–Joyal Semantics: Forcing Over an Arbitrary Category can Always be Replaced by Forcing Over a Complete Heyting Algebra

It is assumed that a Kripke–Joyal semantics ${\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}$ A = C , Cov , F , ? has been defined for a first-order language ${\mathcal{L}}$ L . To transform ${\mathbb{C}}$ C into a Heyting algebra ${\overline{\mathbb{C}}}$ C ¯ on which the forcing relation is preserved, a standard construction is used to obtain a complete Heyting algebra made up of cribles of ${\mathbb{C}}$ C . A pretopology ${\overline{{\rm Cov}}}$ Cov ¯ is defined on ${\overline{\mathbb{C}}}$ C ¯ using the pretopology on ${\mathbb{C}}$ C . A sheaf ${\overline{{\it F}}}$ F ¯ is made up of sections of F that obey functoriality. A forcing relation ${\overline{\Vdash}}$ ? ¯ is defined and it is shown that ${\overline{\mathcal{A}} = \left\langle \overline{\mathbb{C}},\overline{\rm{Cov}},\overline{{\it F}}, \overline{\Vdash} \right\rangle }$ A ¯ = C ¯ , Cov ¯ , F ¯ , ? ¯ is a Kripke–Joyal semantics that faithfully preserves the notion of forcing of ${\mathcal{A}}$ A . That is to say, an object a of ${\mathbb{C}Ob}$ C O b forces a sentence with respect to ${\mathcal{A}}$ A if and only if the maximal a-crible forces it with respect to ${\overline{\mathcal{A}}}$ A ¯ . This reduces a Kripke–Joyal semantics defined over an arbitrary site to a Kripke–Joyal semantics defined over a site which is based on a complete Heyting algebra.     

Exact multiplicity for the perturbed Q-curvature problem in {\mathbb{R}^N,N \geq 5}

Let N ≥ 5 and \({{\mathcal{D}}^{2,2} (\mathbb{R}^N)}\) denote the closure of \({C_0^\infty (\mathbb{R}^N)}\) in the norm \({\|u\|_{{\mathcal{D}}^{2,2} (\mathbb{R}^N)}^2 := \int\nolimits_{\mathbb{R}^N} |\Delta u|^2.}\) Let \({K \in C^2 (\mathbb{R}^N).}\) We consider the following problem for ? ≥ 0: $$(P_\varepsilon) \left\{\begin{array}{llll}{\rm Find} \, u \in {\mathcal{D}}^{2, 2} (\mathbb{R}^N) \, \, {\rm solving} :\\ \left.\begin{array}{lll}\Delta^2 u = (1+ \varepsilon K (x)) u^{\frac{N+4}{N-4}}\\ u > 0 \end{array}\right\}{\rm in} \, \mathbb{R}^N.\end{array}\right.$$ We show an exact multiplicity result for (P _? ) for all small ? > 0.     

p-Laplacian problems with nonlinearities interacting with the spectrum

The aim of this paper is investigating the existence and the multiplicity of weak solutions of the quasilinear elliptic problem $$\left\{\begin{array}{ll}-\Delta_p u\ =\ g(x, u) \quad {\rm in} \quad \Omega,\\ u=0 \qquad \qquad \qquad {\rm on}\quad \partial\Omega,\end{array}\right.$$ where ${1 < p < + \infty, \Delta_p u = {\rm div}(|\nabla {u}|^{p-2}\nabla {u})}$ , Ω is an open bounded domain of ${\mathbb{R}^N (N \geq 3)}$ with smooth boundary ?Ω and the nonlinearity g behaves as u ^p?1 at infinity. The main tools of the proof are some abstract critical point theorems in Bartolo et al. (Nonlinear Anal. 7: 981–1012, 1983), but extended to Banach spaces, and two sequences of quasi–eigenvalues for the p–Laplacian operator as in Candela and Palmieri (Calc. Var. 34: 495–530, 2009), Li and Zhou (J. Lond. Math. Soc. 65: 123–138, 2002).     

Blow-up solutions concentrated along minimal submanifolds for some supercritical elliptic problems on Riemannian manifolds

Let (M, g) and \({(K, \kappa)}\) be two Riemannian manifolds of dimensions m and k, respectively. Let \({\omega \in C^{2} (N), \omega > 0}\) . The warped product \({M \times_\omega K}\) is the (m +  k)-dimensional product manifold \({M \times K}\) furnished with metric \({g + \omega^{2} \kappa}\) . We prove that the supercritical problem $$- \Delta_{g + \omega^{2} \kappa} u + hu = u^{\frac{m+2}{m-2} \pm \varepsilon} ,\quad u > 0,\quad {\rm in}\,\, (M \times_{\omega} K, g + \omega^{2} \kappa)$$ has a solution concentrated along a k-dimensional minimal submanifold \({\Gamma}\) of \({M \times_{\omega } N}\) as the real parameter \({\varepsilon}\) goes to zero, provided the function h and the sectional curvatures along \({\Gamma}\) satisfy a suitable condition.     

Singularly perturbed nonlinear Dirichlet problems involving critical growth

We consider the following singularly perturbed nonlinear elliptic problem: $$\begin{array}{ll}-\varepsilon^{2}\Delta u + u=f(u),\; u > 0\, {\rm on}\, \Omega,\; u = 0\, {\rm on}\, \partial \Omega,\end{array}$$ where Ω is a bounded domain in ${\mathbb{R}^N (N \ge 3)}$ with a boundary ${\partial \Omega \in C^2}$ and the nonlinearity f is of critical growth. In this paper, we construct a solution ${u_\varepsilon}$ of the above problem which exhibits one spike near a maximum point of the distance function from the boundary ?Ω under a critical growth condition on f. Our result complements the study made in [9] in the sense that, in that paper, only the subcritical growth was considered.     

Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C ² boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ ₁ and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\) . We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ?Ω.     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.