This paper is concerned with the 513 isomorphism classes of degree~12 2-adic fields whose automorphism groups have order~4. For each extension, we identify a defining polynomial, the extension's ramification index, residue degree, discriminant, and the Galois group of the extension's normal closure. These results extend previous work of Jones and Roberts, Awtrey and Shill.

We provide explicit expressions for the Liouvillian first integrals of the quadratic polynomial differential systems having an integrable saddle.

Since the 1970's, great interest has been taken in the study of pure $O$-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into two classes of ideals, prove that they are strictly unimodal and show that one of the classes is licci.

With reference to papers of Oikhberg and Rosenthal \cite {OR} and \v {S}emrl \cite {semrl2008, semrl}, we make a contribution to the problem of characterization of metrics generated by norms.

Let $A_{/k}$ denote an abelian variety defined over a number field $k$ with good ordinary reduction at all primes above $p$, and let $K_{\infty }=\bigcup _{n\geq 1} K_n$ be a $p$-adic Lie extension of $k$ containing the cyclotomic $\mathbb{Z}_p$-extension. We use $\mathrm {K}-theory to find recurrence relations for the $\lambda$-invariant at each $\sigma$-component of the Selmer group over $K_{\infty }$, where $\sigma :G_k\rightarrow \mathrm{GL}(V)$. This provides upper bounds on the Mordell-Weil rank for $A(K_n)$ as $n\rightarrow \infty$ whenever $G_{\infty }=\mathrm {Gal}(K_{\infty}/k)$ has dimension at most $3$.

We first introduce in the paper the $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: \[\xymatrix @C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C \ar @\lt 0.5ex>[r]^{C\otimes _R-} \amp \,\,\,\,\mathcal {G}(\mathcal {W}_F) \ar @\lt 0.5ex>[l]^{\textrm {Hom}_R(C,-)}} \] where $\mathcal {G}(\mathcal {F})$, $\mathcal {A}_C$ and $\mathcal {G}(\mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $\mathcal {W}_F$-Gorenstein modules, respectively. Then, we investigate two-degree $\mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $\mathcal {W}_F$-Gorenstein if there exists an exact sequence $\mathbb {G}_\bullet =\cdots \rightarrow G_1\rightarrow G_0\rightarrow G^0\rightarrow G^1\rightarrow \cdots $ in $\mathcal {G}(\mathcal {W}_F)$ such that $M \cong $ $\im \,(G_0\rightarrow G^0) $ and $\mathbb {G}_\bullet $ is $\mbox {Hom}_R(\mathcal {G}(\mathcal {W}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes _R-$ exact. We show that two notions of the two-degree $\mathcal {W}_F$-Gorenstein and the $\mathcal {W}_F$-Gorenstein modules coincide when $R$ is a commutative $GF$-closed ring.

Let $\mathcal {C}_s$ be the set of Dyck paths of length $2s$. For all $s$ its cardinality is given by the Catalan number $C_s$, i.e., $\#\mathcal {C}_s = C_s$. We denote the maximum of a Dyck path $X$ by $X_{\max }$, and thereby we define the exponential moments $E_s(\lambda ):= C^{-1}_s\sum _{X\in \mathcal {C}_s}\exp (\lambda (2s)^{-1/2}X_{\max })$, whereupon we scale the Dyck paths' maxima by the root of the path length. Then, there exists a function $d: \mathbb {R}\to \mathbb {R}$ so that $E_s(\lambda ) \le d(\lambda )$ for all $s\in \mathbb {N}$. We present an elementary proof of this well-known result, which is based on the reflection principle and a telescoping sum trick; in particular, nothing about stochastic processes is used.

We examine the question of direct-sum cancellation of finitely generated, torsion-free modules over ring orders $R$ contained in $\mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$. Using conditions involving the Picard group and the unit group of $R$, we give a nearly complete classification of those orders $R$ for which torsion-free cancellation holds. There is exactly one `exceptional' order to which our methods do not apply.

Let $R$ be a finite commutative ring. In this paper, we find the number of representations of a fixed member of $R$ to be the sum of $k$ units in $R$, and the sum of $k$ non-units, and as a sum of a unit and a non-unit. We prove that, if $\Z _2$ is not a quotient of $R$, then every $r\in R$ can be written as a sum of $k$ units, for each integer $k > 1$.

We consider a nonlocal parabolic and the corresponding elliptic equation with exponential nonlinearity. At first, we study the set of a stationary solution and compute its Morse index. Next, we obtain the time-global solution in the use of the Lyapunov function and define the dynamical system. Finally, we construct an exponential attractor by a squeezing property.

In this paper, we investigate the number of sharing values of a meromorphic function and its derivative in one angular domain instead of the whole complex plane and obtain the following results: Let $f$ be a meromorphic function of lower order $>2$ in the complex plane. Then there exists a direction H: $\arg z=\theta \sb 0$ ($0\leq \theta _0\lt 2\pi $) such that for any positive number $\varepsilon $, $f$ and $f'$ share at most two distinct finite values without counting multiplicities in the angular region $ \{z: |\arg z-\theta _0|\lt \varepsilon \}$. This improve a result of Weichuan and Mori.

Results of Da Prato and Sinestrari \cite {d81} on differential operators with non-dense domain but satisfying the Hille-Yosida condition, are applied in the setting of Beurling weighted spaces of ultradistributions $\DD '^{(s)}_{L^p}((0,T)\times U)$, where $U$ is open and bounded set in $\mathbb R^d$. For this purpose, the new structural theorems were given for $\DD '^{(s)}_{L^p}((0,T)\times U)$. Then a class of the Cauchy problems in the general setting of such spaces of ultradistributions is analyzed.

In this paper, we present extensive numerical data on quadratic function fields with non-zero 3-rank. We use a function field adaptation of a method due to Belabas for finding quadratic number fields of high 3-rank. Our algorithm relies on previous work for tabulating cubic function fields of bounded discriminant \cite {Pieter3} but includes a significant novel improvement when the discriminants are imaginary. We provide numerical data for discriminant degree up to 11 over the finite fields $\mathbb{F}_5, \mathbb{F}_7, \mathbb{F}_11$ and $\mathbb{F}_13$ and $\mathbb{F}_13$. In addition to presenting new examples of fields of minimal discriminant degree with a given 3-rank, we compare our data with a variety of heuristics on the density of such fields with a given 3-rank, which in most cases supports their validity.

In this paper, we introduce a risk model based on the policy entrance process with $n$ kinds of independent policies. Aiming at the model in which each kind of policy is issued according to a non-homogeneous Poisson process with heavy-tailed distributed claim sizes, we study the large deviations for aggregate loss process of the risk model.

Let $\mathscr {A}$ be an abelian category and $\mathscr {X},\mathscr {Y},\mathscr {Z}$ additive full subcategories of $\mathscr {A}$. We introduce and study the Gorenstein category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$ as a common generalization of some known modules such as Gorenstein projective (injective) modules \cite {EJ95}, strongly Gorenstein flat modules \cite {DLM} and Gorenstein FP-injective modules \cite {DM}, and prove the stability of $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$. We also establish Gorenstein homological dimensions in terms of the category $\mathcal {G}(\mathscr {X},\mathscr {Y},\mathscr {Z})$.

In this paper we shall use the generating functions and the double shuffle relations satisfied by the multiple zeta values to derive some new families of identities of these values.