Milnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$ d the singular locus of Md and by ${\mathcal B}$ d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$ 2 is a cubic curve; so ${\mathcal B}$ 2 is connected and ${\mathcal S}$ 2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$ d = ${\mathcal B}$ d . In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$ d .

Using a variational approach we obtain the existence of at least three periodic solutions for discontinuous perturbations of the vector p-Laplacian operator .

We consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn , 0) and a function germ f : (ℂn , 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

We provide ℕ-filtrations on the negative part Uq ( $\mathfrak{n}$ −) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$ , such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$ −. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$ -module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$ .

We show that every fibrewise map from a Serre microfibration to a Serre fibration is n-connected if it is fibrewise n-connected. This generalises a result of M. Weiss and related results by Bökstedt–Madsen and Galatius–Randal–Williams. We also discuss an application to configuration spaces.

In this paper, we provide an explicit upper bound for the Loxton–Kedlaya rank of the maximal abelian extension of ℚ.

In a Markov chain started at a state x, the hitting time τ(y) is the first time that the chain reaches another state y. We study the probability $\mathbb{P}_x(\tau(y) = t)$ that the first visit to y occurs precisely at a given time t. Informally speaking, the event that a new state is visited at a large time t may be considered a ‘surprise’. We prove the following three bounds.

• • In any Markov chain with n states, $\mathbb{P}_x(\tau(y) = t) \le {n}/{t}$ .

• • In a reversible chain with n states, $\mathbb{P}_x(\tau(y) = t) \le {\sqrt{2n}}/{t}$ for $t \ge 4n + 4$ .

• • For random walk on a simple graph with n ≥ 2 vertices, $\mathbb{P}_x(\tau(y) = t) \le 4e \log(n)/t$ .

To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): for random walk on an n-vertex graph, for every initial vertex x,

$$\sum_y \biggl( \sup_{t \ge 0} p^t(x, y) \biggr) = O(\log n). $$

We deal with the problem of determining an unknown part of the boundary of an electrical conductor that is inaccessible for external observation and where a corrosion process is going on. We obtain estimates of the size of this damaged region from above and below.

In this paper we generalize the notion of the C-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator C, the C-numerical range of such an operator is defined; it is a compact, convex subset of ℂ. We explicitly describe the C-numerical ranges of several operators and classes of operators.

We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus conjecture for the group PSL(2, 19). We also prove the Zassenhaus conjecture for PSL(2, 23). In a second application we show that there are no normalized units of order 6 in the integral group rings of M 10 and PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

We employ some formulae on hypergeometric series and p-adic Gamma function to establish several congruences.

This paper is concerned with the construction of global, non-vacuum, strong, large amplitude solutions to initial–boundary-value problems for the one-dimensional compressible Navier–Stokes equations with degenerate transport coefficients. Our analysis derives the positive lower and upper bounds on the specific volume and the absolute temperature.

We consider a fractional equation involving the left and right Riemann–Liouville fractional integrals and with Sturm–Liouville boundary-value conditions. We establish the variational structure of the problem and, by using critical-point theory, the existence of an unbounded sequence of solutions is obtained.

A well-posedness theory for the initial-value problem for hydroelastic waves in two spatial dimensions is presented. This problem, which arises in numerous applications, describes the evolution of a thin elastic membrane in a two-dimensional (2D) potential flow. We use a model for the elastic sheet that accounts for bending stresses and membrane tension, but which neglects the mass of the membrane. The analysis is based on a vortex sheet formulation and, following earlier analyses and numerical computations in 2D interfacial flow with surface tension, we use an angle–arclength representation of the problem. We prove short-time well-posedness in Sobolev spaces. The proof is based on energy estimates, and the main challenge is to find a definition of the energy and estimates on high-order non-local terms so that an a priori bound can be obtained.

We study Lotka–Volterra models with fractional Laplacian. To do this we study in detail the logistic problem and show that the sub–supersolution method works for both the scalar problem and for systems. We apply this method to show the existence and non-existence of positive solutions in terms of the system parameters.

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GLn (q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n 2(1 – a −1) − εr and leading coefficient mr , where εr and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

The study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules.

Let $\mathbb{K}$ be a field and S = ${\mathbb{K}}$ [x 1, . . ., xn ] be the polynomial ring in n variables over the field $\mathbb{K}$ . For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g., a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundles over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated with parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra.

In this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H 1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.