Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

Convergence of symmetric Markov chains on {\mathbb{Z}^d}

For each n let ${Y^{(n)}_t}$ be a continuous time symmetric Markov chain with state space ${n^{-1} \mathbb{Z}^d}$ . Conditions in terms of the conductances are given for the convergence of the ${Y^{(n)}_t}$ to a symmetric Markov process Y _t on ${\mathbb{R}^d}$ . We have weak convergence of $\{{Y^{(n)}_t: t \leq t_0\}}$ for every t ₀ and every starting point. The limit process Y has a continuous part and may also have jumps.     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Duality of weighted anisotropic Besov and Triebel–Lizorkin spaces

Let A be an expansive dilation on ${{\mathbb R}^n}$ and w a Muckenhoupt ${\mathcal A_\infty(A)}$ weight. In this paper, for all parameters ${\alpha\in{\mathbb R} }$ and ${p,q\in(0,\infty)}$ , the authors identify the dual spaces of weighted anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A;w)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A;w)}$ with some new weighted Besov-type and Triebel?CLizorkin-type spaces. The corresponding results on anisotropic Besov spaces ${\dot B^\alpha_{p,q}(A; \mu)}$ and Triebel?CLizorkin spaces ${\dot F^\alpha_{p,q}(A; \mu)}$ associated with ${\rho_A}$ -doubling measure??? are also established. All results are new even for the classical weighted Besov and Triebel?CLizorkin spaces in the isotropic setting. In particular, the authors also obtain the ${\varphi}$ -transform characterization of the dual spaces of the classical weighted Hardy spaces on ${{\mathbb R}^n}$ .     

A sharp lower bound for the log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function ${\varphi}$ with an isolated singularity at 0 in an open subset of ${\mathbb{C}^n}$ . This threshold is defined as the supremum of constants c > 0 such that ${e^{-2c\varphi}}$ is integrable on a neighborhood of 0. We relate ${c(\varphi)}$ to the intermediate multiplicity numbers ${e_j(\varphi)}$ , defined as the Lelong numbers of ${(dd^c\varphi)^j}$ at 0 (so that in particular ${e_0(\varphi)=1}$ ). Our main result is that ${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$ . This inequality is shown to be sharp; it simultaneously improves the classical result ${c(\varphi)\geqslant 1/e_1(\varphi)}$ due to Skoda, as well as the lower estimate ${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.     

Cluster Algebras of Type A _{\bf{2}}^{\bf {(1)}}

In this paper we study cluster algebras $\mathcal{A}$ of type $A_2^{(1)}$ . We solve the recurrence relations among the cluster variables (which form a T-system of type $A_2^{(1)}$ ). We solve the recurrence relations among the coefficients of $\mathcal{A}$ (which form a Y-system of type $A_2^{(1)}$ ). In $\mathcal{A}$ there is a natural notion of positivity. We find linear bases B of $\mathcal{A}$ such that positive linear combinations of elements of B coincide with the cone of positive elements. We call these bases atomic bases of $\mathcal{A}$ . These are the analogue of the “canonical bases” found by Sherman and Zelevinsky in type $A_{1}^{(1)}$ . Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of B are parameterized by ?³ via their g-vectors in every cluster. We prove that the denominator vector map in every acyclic seed of $\mathcal{A}$ restricts to a bijection between B and ?³. We find explicit recurrence relations to express every element of $\mathcal{A}$ as linear combinations of elements of B.     

On Suborbital Graphs for the Extended Modular Group {\hat{\Gamma}}

In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$      

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

SIP: critical value functions have finite modulus of non-convexity

We consider semi-infinite programming problems ${{\rm SIP}(z)}$ depending on a finite dimensional parameter ${z \in \mathbb{R}^p}$ . Provided that ${\bar{x}}$ is a strongly stable stationary point of ${{\rm SIP}(\bar{z})}$ , there exists a locally unique and continuous stationary point mapping ${z \mapsto x(z)}$ . This defines the local critical value function ${\varphi(z) := f(x(z); z)}$ , where ${x \mapsto f(x; z)}$ denotes the objective function of ${{\rm SIP}(z)}$ for a given parameter vector ${z\in \mathbb{R}^p}$ . We show that ${\varphi}$ is the sum of a convex function and a smooth function. In particular, this excludes the appearance of negative kinks in the graph of ${\varphi}$ .     

BIB-Designs from Circular Nearrings

Let ${(N, \Phi)}$ be a finite circular Ferrero pair. We define the disk with center b and radius ${a, \mathcal{D}(a;b)}$ , as $$\mathcal{D} (a; b) = \{x \in \Phi(r)+c \mid r \neq 0, b\in \Phi (r)+c, |(\Phi (r)+c) \cap ( \Phi(a)+b)|=1\}.$$ Using this definition we introduce the concept of interior part of a circle, ${\Phi(a)+b}$ , as the set ${\mathcal{I}(\Phi (a)+b)=\mathcal{D} (a; b) \setminus (\Phi (a)+b)}$ . Moreover, if ${\mathcal{B}^{\mathcal{D}}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure ${(N, \mathcal{B}^{\mathcal{D}}, \in)}$ is actually a balanced incomplete block design and we are able to calculate its parameters depending on |N| and ${|\Phi|}$ .     

On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols

Let ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ denote the standard weighted Bergman space over the unit ball ${\mathbb{B}^n}$ in ${\mathbb{C}^n}$ . New classes of commutative Banach algebras ${\mathcal{T}(\lambda)}$ which are generated by Toeplitz operators on ${\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}$ have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141?C152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of ${\mathbb{B}^n}$ . In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n?=?2. We explicitly describe the maximal ideal space and the Gelfand map of ${\mathcal{T}(\lambda)}$ . Since ${\mathcal{T}(\lambda)}$ is not invariant under the *-operation of ${\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}$ its inverse closedness is not obvious and is proved. We remark that the algebra ${\mathcal{T}(\lambda)}$ is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in ${\mathcal{T}(\lambda)}$ is always connected.     

Planar Beurling Transform and Bergman Type Spaces

In this paper we describe the actions of the operator $S_\mathbb{D }$ or its adjoint $S_\mathbb{D }^*$ on the poly-Bergman spaces of the unit disk $\mathbb{D }.$ Let $k$ and $j$ be positive integers. We prove that $(S_\mathbb{D })^{j}$ is an isometric isomorphism between the true poly-Bergman subspace $\mathcal{A }_{(k)}^2(\mathbb{D })\ominus N_{(k),j}$ onto the true poly-Bergman space $\mathcal{A }_{(j+k)}^2(\mathbb{D }),$ where the linear space $N_{(k),j}$ have finite dimension $j.$ The action of $(S_\mathbb{D })^{j-1}$ on the canonical Hilbert base for the Bergman subspace $\mathcal{A }^2(\mathbb{D })\ominus \mathcal{P }_{j-1},$ gives a Hilbert base $\{ \phi _{ j , k } \}_{ k }$ for $\mathcal{A }_{(j)}^2(\mathbb{D }).$ It is shown that $\{ \phi _{ j , k } \}_{ j, k }$ is a Hilbert base for $L^2(\mathbb{D },d A)$ such that whenever $j$ and $k$ remain constant we obtain a Hilbert base for the true poly-Bergman space $\mathcal{A }_{(j)}^2(\mathbb{D })$ and $\mathcal{A }_{(-k)}^2(\mathbb{D }),$ respectively. The functions $\phi _{ j , k }$ are polynomials in $z$ and $\overline{z}$ and are explicitly given in terms of the $(2,1)$ -hypergeometric polynomials. We prove explicit representations for the true poly-Bergman kernels and the Koshelev representation for the poly-Bergman kernels of $\mathbb{D }.$ The action of $S_\Pi $ on the true poly-Bergman spaces of the upper half-plane $\Pi $ allows one to introduce Hilbert bases for the true poly-Bergman spaces, and to give explicit representations of the true poly-Bergman and poly-Bergman kernels.     

A hybrid inequality of Erdös–Turán–Koksma for digital sequences

For bases $\mathbf{b}=(b_1, \ldots , b_s)$ of $s$ not necessarily distinct integers $b_i\ge 2$ , we prove a version of the inequality of Erdös–Turán–Koksma for the hybrid function system composed of the Walsh functions in base $\mathbf{b}^{(1)}=(b_1, \ldots , b_{s_1})$ and, as second component, the $\mathbf{b}^{(2)}$ -adic functions, $\mathbf{b}^{(2)}=(b_{s_1+1}, \ldots , b_s)$ , with $s=s_1+s_2$ , $s_1$ and $s_2$ not both equal to 0. Further, we point out why this choice of a hybrid function system covers all possible cases of sequences that employ addition of digit vectors as their main construction principle.     

Fermat-type equations of signature (13,13,p) via Hilbert cuspforms

In this paper we prove that for $p > 13649$ equations of the form $x^{13} + y^{13} = Cz^{p}$ have no non-trivial primitive solutions $(a,b,c)$ such that $13 \not \mid c$ for an infinite family of values for $C$ . Our method consists on relating a solution $(a,b,c)$ to the previous equation to a solution $(a,b,c_1)$ of another Diophantine equation with coefficients in $\mathbb Q (\sqrt{13})$ . Then we attach to $(a,b,c_1)$ a Frey curve $E_{(a,b)}$ defined over $\mathbb Q (\sqrt{13})$ that is not a $\mathbb Q $ -curve. We prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at $3$ . This theorem, in particular, implies modularity of $E_{(a,b)}$ . This enables us to use level lowering results and apply the modular approach via Hilbert cuspforms over $\mathbb Q (\sqrt{13})$ to prove the non-existence of $(a,b,c_1)$ and, consequently, of $(a,b,c)$ .     

Compact composition operators acting between weighted Bergman spaces of the unit ball

In this paper, we study a composition operator ${C_{\varphi}}$ on the weighted Bergman space ${A_{\alpha}^p(B)}$ of the unit ball B in ${{\mathbb{C}}^N}$ . Under a natural condition we estimate the essential norm of ${C_{\varphi}}$ . As a consequence of this estimate, we also give a function-theoretic characterization of ${\varphi}$ that induces a compact composition operator on ${A_{\alpha}^p(B)}$ .     

Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process

We consider a discrete-time two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ on $\mathbb{Z}_{+}^{2}$ with a background process {J _n } on a finite set, where individual processes $\{L_{n}^{(1)}\}$ and $\{L_{n}^{(2)}\}$ are both skip free. We assume that the joint process $\{Y_{n}\}=\{(L_{n}^{(1)},L_{n}^{(2)},J_{n})\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ are modulated depending on the state of the background process {J _n }. This modulation is space homogeneous, but the transition probabilities in the inside of $\mathbb{Z}_{+}^{2}$ and those around the boundary faces may be different. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process, and obtain the decay rates of the stationary distribution in the coordinate directions. We also distinguish the case where the stationary distribution asymptotically decays in the exact geometric form, in the coordinate directions.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.