Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

The rearrangement-invariant space \Gamma _{p,\phi }

Fix \(b\in \mathbb R _+\) and \(p\in (1,\infty )\) . Let \(\phi \) be a positive measurable function on \(I_b:=(0,b)\) . Define the Lorentz Gamma norm, \(\rho _{p,\phi }\) , at the measurable function \(f:\mathbb R _+\rightarrow \mathbb R _+\) by \(\rho _{{}_{p,\phi }}(f):=\left[ \int _0^bf^{**}(t)^p\phi (t)\,dt\right] ^{\frac{1}{p}}\) , in which \(f^{**}(t):=t^{-1}\int _0^tf^{*}(s)\,ds\) , where \(f^*(t):=\mu _f^{-1}(t)\) , with \(\mu _f(s):=|\{ x\in I_b: |f(x)|>s\}|\) . Our aim in this paper is to study the rearrangement-invariant space determined by \(\rho _{{}_{p,\phi }}\) . In particular, we determine its Köthe dual and its Boyd indices. Using the latter a sufficient condition is given for a Caldéron–Zygmund operator to map such a space into itself.     

A difference version of Nori’s theorem

We consider (Frobenius) difference equations over \((\mathbb {F}\!_q(s,t), \phi _q)\) where \(\phi _q\) fixes \(t\) and acts on \(\mathbb {F}\!_q(s)\) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group \(\mathcal {G}\) defined over \(\mathbb {F}\!_q\) can be realized as a difference Galois group over \((\mathbb {F} \! _{q^i} (s,t),\phi _{q^i})\) for some \(i \in \mathbb {N}\) . The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori’s theorem which states that \(\mathcal {G}(\mathbb {F}\!_q)\) occurs as a (finite) Galois group over \(\mathbb {F}\!_q(s)\) .     

On Artin Algebras Arising from Morita Contexts

We study Morita rings \(\Lambda _{(\phi ,\psi )}=\left (\begin {array}{cc}A &_{A}N_{B} \\ _{B}M_{A} & B \end {array}\right )\) in the context of Artin algebras from various perspectives. First we study covariantly finite, contravariantly finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms \(\phi \) and \(\psi \) are zero. Further we give bounds for the global dimension of a Morita ring \(\Lambda _{(0,0)}\) , as an Artin algebra, in terms of the global dimensions of A and B in the case when both \(\phi \) and \(\psi \) are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring \(\Lambda _{\phi ,\psi }\) in case \(A=N=M=B\) and A an Artin algebra.     

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\) , in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\) , where \(r_i\) is either \(1\) or \(2\) .     

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation \(T\) of a probability space \((X,\mathcal{F },\mu )\) and some \(d \ge 1\) we investigate almost sure and distributional convergence of random variables of the form $$\begin{aligned} x \rightarrow \frac{1}{C_n} \sum _{0\le i_1,\ldots ,\,i_d where \(C_1, C_2,\ldots \) are normalizing constants and the kernel \(f\) belongs to an appropriate subspace in some \(L_p(X^d\!,\, \mathcal{F }^{\otimes d}\!,\,\mu ^d)\) . We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with \(T\) and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and \(d=2\) , we also show that the convergence holds in distribution towards a quadratic form \(\sum _{m=1}^{\infty } \lambda _m\eta ^2_m\) in independent standard Gaussian variables \(\eta _1, \eta _2, \ldots \) .     

The theme of a vanishing period

Consider a multivalued formal function of the type 1 $$\begin{aligned} \varphi (s) : = \sum _{j=0}^k\,c_j(s).s^{\lambda + m_j}.(\mathrm{Log}\,s)^j, \end{aligned}$$ where \(\lambda \) is a positive rational number, \(c_j\) is in \({{\mathrm{\mathbb {C}}}}[[s]]\) and \(m_j \in \mathbb {N}\) for \(j \in [0,k-1]\) . The theme associated with such a \(\varphi \) is the “minimal filtered integral equation” satisfied by \(\varphi \) , in a sense which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial of \(\varphi \) to be fixed. For a given \(\lambda \) , to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated with the logarithm of the monodromy in the geometric situation described below. Our aim is to construct some analytic invariants, for instance in the following situation, let \(f : X \rightarrow D\) be a proper holomorphic function defined on a complex manifold \(X\) with values in a disc \(D\) . We assume that the only critical value is \(0 \in D\) and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \(f^{-1}(0)\) . To a smooth \((p+1)\) -form \(\omega \) on \(X\) such that \(\mathrm{d}\omega = 0 = \mathrm{d}f \wedge \omega \) and to a vanishing \(p\) -cycle \(\gamma \) chosen in the generic fiber \(f^{-1}(s_0), s_0 \in D \setminus \{0\}\) , we associated a “vanishing period” \(F_{\gamma }(s) : = \int _{\gamma _s} \omega \big /\mathrm{d}f \) which has an asymptotic expansion at \(0\) of the form \((1)\) above, when \(\gamma \) is chosen in the spectral subspace of \(H_p(f^{-1}(s_0), {{\mathrm{\mathbb {C}}}})\) for the eigenvalue \(\mathrm{e}^{2i\pi .\lambda }\) of the monodromy of \(f\) . Here \((\gamma _s)_{s \in D^*}\) is the horizontal multivalued family of \(p\) -cycles in the fibers of \(f\) obtained from the choice of \(\gamma \) . The aim of this article was to study the module generated by such a \(\varphi \) over the algebra \(\tilde{\mathcal {A}}\) , which is the \(b\) -completion of the algebra \(\mathcal {A}\) generated by the operators \(\mathrm{a} : = \times s\) and \(\mathrm{b} : = \int _{0}^{s}\) .     

Mean Squared Error Minimization for Inverse Moment Problems

We consider the problem of approximating the unknown density \(u\in L^2(\Omega ,\lambda )\) of a measure \(\mu \) on \(\Omega \subset \mathbb {R}^n\) , absolutely continuous with respect to some given reference measure \(\lambda \) , only from the knowledge of finitely many moments of \(\mu \) . Given \(d\in \mathbb {N}\) and moments of order \(d\) , we provide a polynomial \(p_d\) which minimizes the mean square error \(\int (u-p)^2d\lambda \) over all polynomials \(p\) of degree at most \(d\) . If there is no additional requirement, \(p_d\) is obtained as solution of a linear system. In addition, if \(p_d\) is expressed in the basis of polynomials that are orthonormal with respect to \(\lambda \) , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover \(p_d\rightarrow u\) in \(L^2(\Omega ,\lambda )\) as \(d\rightarrow \infty \) . In general nonnegativity of \(p_d\) is not guaranteed even though \(u\) is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing \(p_d\ge 0\) that minimizes \(\int (u-p)^2d\lambda \) now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating \(u\) .     

Estimates for vector valued Dirichlet polynomials

We estimate the \(\ell _1\) -norm \(\sum _{n=1}^N \Vert a_n\Vert \) of finite Dirichlet polynomials \(\sum _{n=1}^N a_n n^{-s},\,s \in {\mathbb {C}}\) with coefficients \(a_n\) in a Banach space. Our estimates quantify several recent results on Bohr’s strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.     

Carleson Type Measures for Fock–Sobolev Spaces

We describe the \((p,q)\) Fock–Carleson measures for weighted Fock–Sobolev spaces in terms of the objects \((s,t)\) -Berezin transforms, averaging functions, and averaging sequences on the complex space \(\mathbb{C }^n\) . The main results show that while these objects may have growth not faster than polynomials to induce the \((p,q)\) measures for \(q\ge p\) , they should be of \(L^{p/(p-q)}\) integrable against a weight of polynomial growth for \(q<p\) . As an application, we characterize the bounded and compact weighted composition operators on the Fock–Sobolev spaces in terms of certain Berezin type integral transforms on \(\mathbb{C }^n\) . We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.     

Spherical functions associated with the three-dimensional sphere

In this paper, we determine all irreducible spherical functions \(\Phi \) of any \(K \) -type associated with the pair \((G,K)=(\mathrm{SO }(4),\mathrm{SO }(3))\) . This is accomplished by associating with \(\Phi \) a vector-valued function \(H=H(u)\) of a real variable \(u\) , which is analytic at \(u=0\) and whose components are solutions of two coupled systems of ordinary differential equations. By an appropriate conjugation involving Hahn polynomials, we uncouple one of the systems. Then, this is taken to an uncoupled system of hypergeometric equations, leading to a vector-valued solution \(P=P(u)\) , whose entries are Gegenbauer’s polynomials. Afterward, we identify those simultaneous solutions and use the representation theory of \(\mathrm{SO }(4)\) to characterize all irreducible spherical functions. The functions \(P=P(u)\) corresponding to the irreducible spherical functions of a fixed \(K\) -type \(\pi _\ell \) are appropriately packaged into a sequence of matrix-valued polynomials \((P_w)_{w\ge 0}\) of size \((\ell +1)\times (\ell +1)\) . Finally, we prove that \(\widetilde{P}_w={P_0}^{-1}P_w\) is a sequence of matrix orthogonal polynomials with respect to a weight matrix \(W\) . Moreover, we show that \(W\) admits a second-order symmetric hypergeometric operator \(\widetilde{D}\) and a first-order symmetric differential operator \(\widetilde{E}\) .     

Strongly amenable involutive representations of involutive Banach algebras

Let \(\mathfrak{A }\) be a Banach \(*\) -algebra and let \(\varphi \) be a nonzero self-adjoint character on \(\mathfrak{A }\) . For a   \(*\) -representation \(\pi \) of \(\mathfrak{A }\) on a Hilbert space \(\mathcal{H }\) , we introduce and study strong \(\varphi \) -amenability of \(\pi \) in terms of certain states on the von Neumann algebra of bounded operators on \(\mathcal{H }\) . We then give some characterizations of this notion in terms of certain positive functionals on \(\mathfrak{A }\) . We finally investigate some hereditary properties of strong \(\varphi \) -amenability of Banach algebras.     

A class of \ell ^p saturated Banach spaces

We present a new class of reflexive \(\ell ^p\) saturated Banach spaces \(\mathfrak{X }_p\) for \(1<p<\infty \) with rather tight structure. The norms of these spaces are defined with the use of a modification of the standard method yielding hereditarily indecomposable Banach spaces. The space \(\mathfrak{X }_p\) does not embed into a space with an unconditional basis and for any analytic decomposition into two subspaces, it is proved that one of them embeds isomorphically into the \(\ell ^p\) -sum of a sequence of finite dimensional normed spaces. We also study the space of operators of \(\mathfrak{X }_p\) .     

Hall Polynomials and Composition Algebra of Representation Finite Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k such that the indecomposable \(\mathcal{A}\) -modules are determined by their dimension vectors and for each \(M, L \in ind(\mathcal{A})\) and \(N\in mod(\mathcal{A})\) , either \(F^{M}_{N L}=0\) or \(F^{M}_{L N}=0\) . We show that \(\mathcal{A}\) has Hall polynomials and the rational extension of its Ringel–Hall algebra equals the rational extension of its composition algebra. This result extend and unify some known results about Hall polynomials. As a consequence we show that if \(\mathcal{A}\) is a representation finite simply-connected algebra, or finite dimensional k-algebra such that there are no short cycles in \(mod(\mathcal{A})\) , or representation finite cluster tilted algebra, then \(\mathcal{A}\) has Hall polynomials and \(\mathcal{H}(\mathcal{A})\otimes_\mathbb{Z}Q=\mathcal{C}(\mathcal{A})\otimes_\mathbb{Z}Q\) .     

Ribbon graphs and mirror symmetry

Given a ribbon graph \(\Gamma \) with some extra structure, we define, using constructible sheaves, a dg category \(\mathrm {CPM}(\Gamma )\) meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by \(\Gamma .\) When \(\Gamma \) is appropriately decorated and admits a combinatorial “torus fibration with section,” we construct from \(\Gamma \) a one-dimensional algebraic stack \(\widetilde{X}_\Gamma \) with toric components. We prove that our model is equivalent to \(\mathcal {P}\mathrm {erf}(\widetilde{X}_\Gamma )\) , the dg category of perfect complexes on \(\widetilde{X}_\Gamma \) .     

On semi-classical limits of ground states of a nonlinear Maxwell–Dirac system

We study the semi-classical ground states of the nonlinear Maxwell–Dirac system: $$\begin{aligned} \left\{ \begin{array}{l} \alpha \cdot \left( i\hbar \nabla + q(x)\mathbf{A }(x)\right) w-a\beta w -\omega w - q(x)\phi (x) w = P(x)g(\left| w\right| ) w\\ -\Delta \phi =q(x)\left| w\right| ^2\\ -\Delta {A_k}=q(x)(\alpha _k w)\cdot \bar{w}\ \ \ \ k=1,2,3 \end{array} \right. \end{aligned}$$ for \(x\in \mathbb{R }^3\) , where \(\mathbf{A }\) is the magnetic field, \(\phi \) is the electron field and \(q\) describes the changing pointwise charge distribution. We develop a variational method to establish the existence of least energy solutions for \(\hbar \) small. We also describe the concentration behavior of the solutions as \(\hbar \rightarrow 0\) .     

Polynomials for \mathrm{GL}_p\times \mathrm{GL}_q orbit closures in the flag variety

The subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\) -orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\) -polynomials and the Kazhdan–Lusztig–Vogan polynomials.     

On canonical metrics on Cartan–Hartogs domains

The Cartan–Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan–Hartogs domain \(\Omega ^{B^{d_0}}(\mu )\) endowed with the canonical metric \(g(\mu ),\) we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space \(\mathcal {H}_{\alpha }\) of square integrable holomorphic functions on \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) with the weight \(\exp \{-\alpha \varphi \}\) (where \(\varphi \) is a globally defined Kähler potential for \(g(\mu )\) ) for \(\alpha >0\) , and, furthermore, we give an explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion for \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .\) Secondly, using the explicit expression of the Rawnsley’s \(\varepsilon \) -function expansion, we show that the coefficient \(a_2\) of the Rawnsley’s \(\varepsilon \) -function expansion for the Cartan–Hartogs domain \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is constant on \(\Omega ^{B^{d_0}}(\mu )\) if and only if \(\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) \) is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.     

Pro-\mathcal {C} completions of orientable \text{ PD }^3-pairs

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro- \({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\) -pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\) -pair over \(\mathbb {F}_p\) . More results are proven for the pro- \(p\) completion of \(\text{ PD }^3\) -pairs.