We construct explicit examples of non-trivial nilpotent Toeplitz operators on Bergman spaces of certain Reinhardt domains in $\mathbb {C}^2$.

The smallest non-abelian $p$-groups play a fundamental role in the theory of Galois $p$-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups--as well as other closely related, larger $p$-groups--occur as Galois groups over given base fields. We show further how the appearance of some Galois groups forces the appearance of other Galois groups in an interesting way.

We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of intersections. We also associate a topological space: the complement of the union of tangent bundles of these subspheres in the tangent bundle of the ambient sphere. We call this space the tangent bundle complement. As in the case of hyperplane arrangements the aim of this new notion is to understand the interaction between the combinatorics of the intersections and the topology of the tangent bundle complement. In the present paper, we find a closed form formula for the homotopy type of the complement and express some of its topological invariants in terms of the associated combinatorial information.

A quotient ring of a regular local ring of embedding dimension~3 has a minimal free resolution that carries a differential graded algebra structure. This structure has been used to classify such rings. For some classes very few examples have been observed. One such class is called \lt B>{B}\lt /B>, named for Anne Brown, who discovered and gave the first constructions of rings in this class. Her constructions were the only known rings in that class until recently. In this paper, we show how to obtain Artinian rings in the class \textbf {B} by expanding the socle of a complete intersection.

Mixed chord-integrals of star bodies were first defined by Lu. In this paper, the concept of mixed chord-integrals is extended to general mixed chord-integrals, which is motivated by the recent work on general $L_p$-affine isoperimetric inequalities by Haberl, et al. For this new notion of general mixed chord-integrals, isoperimetric and Aleksandrov-Fenchel type inequalities are established which generalize inequalities obtained by Lu, and a cyclic inequality is also obtained. Furthermore, we prove several Brunn-Minkowski type inequalities for general mixed chord-integrals.

We will study a class of nonlinear elliptic systems involving Steklov-Neumann boundary conditions. We obtain results ensuring the existence of solutions when resonance and nonresonance conditions occur. The results were obtained by using variational arguments.

In this paper, we will compute Buchsbaum-Rim functions of two variables associated to a parameter matrix of a special form over a one-dimensional Cohen-Macaulay local ring, and we will determine when the function coincides with the Buchsbaum-Rim polynomial. As a consequence, we have that there exists the case where the function does not coincide with the polynomial function, which should be contrasted with the ordinary Buchsbaum-Rim function of single variable.

Let $R$ be any subring of the reals. We present a generalization of linear systems on graphs where divisors are $R$-valued functions on the set of vertices and graph edges are permitted to have nonnegative weights in $R$. Using this generalization, we provide an independent proof of a Riemann-Roch formula, which implies the Riemann-Roch formula of Baker and Norine.

A proof of the Greene-Krantz conjecture on convex domains in $\mathbb{C} ^2$ is given. Curiously, the proof technique depends on subelliptic estimates for the $\overline {\partial }$ problem.

We use elementary combinatorial methods, together with the theory of quadratic forms, over finite fields to obtain the formula, originally due to Kuz'min, for the number of monic irreducible polynomials of degree~$n$ over a finite field $\mathbb{F} _q$ with the first two prescribed coefficients. The formula relates the number of such irreducible polynomials to the number of polynomials that split over the base field.

We classify all of the weight-homogeneous planar polynomial differential systems of weight degree~$4$ having a polynomial first integral.

We show that a one-parameter unfolding $ F:(\mathbb {R}^3 \times \mathbb {R}, 0) \rightarrow (\mathbb {R}^3 \times \mathbb {R}, 0) $ of a finitely determined map germ $f$, with $S(f)$ regular, is topologically trivial if it is excellent in the sense of Gaffney, and the family of the double point curves and cuspidal edges $D(f_t) \cup C(f_t)$ is topologically trivial.

In this paper, we derive a class of approximations of the $q$-digamma function $\psi _q(x)$. The infinite family \[ I_a(x;q)=\log [x+a]_q+\frac {q^x\log q}{1-q^x}-\bigg (\frac 12-a\bigg )H(q-1)\log q, \] $a\in [0,1]$; $q>0$, can be used as approximating functions for $\psi _q(x)$, where $[x]_q=(1-q^x)/(1-q)$ and $H(\cdot )$ is the Heaviside step function. We show that, for all $a\in [0,1]$, $I_a$ is asymptotically equivalent to $\psi _q(x)$ for $q>0$ and is a good pointwise approximation.

By using the critical point theory, the existence of periodic solutions for 2$n$th-order nonlinear $p$-Laplacian difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point theorem. The problem is to solve the existence of periodic solutions for 2$n$th-order $p$-Laplacian difference equations. The results obtained successfully generalize and complement the existing ones.

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann $\zeta $ function, in particular, the zeros of this function. Theorem~1.2 gives a zero-free region. Theorem~1.4 gives an asymptotic estimate for the number of nontrivial zeros to height $T$. Theorem~1.7 is a zero density estimate.

The existence of affine-periodic solutions is studied. These types of solutions may be periodic, harmonic or even quasi-periodic. Mainly, via the topological degree theory, a general existence theorem is proved, which asserts the existence of affine-periodic solutions, extending some classical results. The theorem is applied to establish the Lyapunov function type theorem and the invariant region principle relative to affine-periodic solutions.

In this paper, we first investigate the relationship between $\mathcal{W} $-(co)resolutions and $\mathcal{X} $-(co)resolutions for two full subcategories $\mathcal{W} $ and $\mathcal{X} $ of an abelian category with $\mathcal{W} \subseteq \mathcal{X} $. Then some applications are given. In particular, we obtain the stability of the category of $C$-Gorenstein flat modules under the procedure used to define these entities, which is different from that established by Sather-Wagstaff, Sharif and White.