For $C^1$ diffeomorphisms of three dimensional closed manifolds, we provide a geometric model of mixing Lyapunov exponents inside a homoclinic class of a periodic saddle $p$ with non-real eigenvalues. Suppose $p$ has stable index two and the sum of the largest two Lyapunov exponents is greater than $\log(1-\delta)$, then $\delta$-weak contracting eigenvalues are obtained by an arbitrarily small $C^1$ perturbation. Using this result, we give a sufficient condition for stabilizing a homoclinic tangency within a given $C^1$ perturbation range.

We prove the analogue of the Doeblin-Lenstra conjecture for a certain complex continued fraction algorithm introduced by Julius Hurwitz [5] in 1895 and further investigated by Shigeru Tanaka [12] in 1985 using ergodic theoretical means.

On some connected sums of 4-manifolds with natural actions of finite groups, we use equivariant Bauer-Furuta invariant to deduce the existence of solutions of Seiberg-Witten equations invariant under the group actions. For example, for any integer $k\geq 2$ we show that the connected sum of $k$ copies of a 4-manifold $M$ with nontrivial Bauer-Furuta invariant has a nontrivial $\mathbb{Z}_k$-equivariant Bauer-Furuta invariant for the obviously glued Spin$^c$ structure, where the $\mathbb{Z}_k$-action cyclically permutes $k$ summands of $M$. This contrasts with the fact that ordinary Bauer-Furuta invariants of such connected sums are all trivial for any sufficiently large $k$, when $b_1(M)=0$.

The homological monodromy of a degeneration whose singular fiber has at most normal crossings was described by C. H. Clemens. In his work, local monodromies were described in detail. It is actually a classical result that the local monodromy around a node is a Dehn twist. For higher-dimensional case, we describe local monodromies alternatively: On a local smooth fiber of dimension $n \geq 2$, we construct $n+1$ singular foliations and then describe the action of the local monodromy on each leaf. Here the $i$th singular foliation is used for describing its action on the $i$th face of the boundary of a local smooth fiber.

Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$. The purpose of this paper is to show that all homomorphic images of the $R$-modules $\mathrm{Ext}^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$, whenever $\dim R \leq4$ or $\dim R/ \frak a \leq 3$ and $R$ contains a field. In addition, we show that if $\dim R=5$ and $R$ contains a field, then the $R$-modules $\mathrm{Ext}^j_R(N, H^i_{\frak a}(R))$ have only finitely many associated primes, for all $i, j\geq 0$.

In this paper, we introduce the concept of $(\psi ,\varphi)_{s}$-weakly $C$-contractive mappings in the setup of partially ordered $b$-metric spaces and investigate some fixed point and common fixed point results for such mappings. Our main results generalize several well-known comparable results in the recent literature. Furthermore, we furnish some suitable examples and an applications of a common solution for a system of integral equations to illustrate the effectiveness and usability of our obtained results.

We apply recent methods of Burns, Kurihara and Sano in [5] to study connections between the values at \(s=0\) of the higher derivatives of abelian \(L\)-functions of number fields and the higher Fitting ideals of the canonical Selmer groups of \(\mathbb{G}_m\). Whereas Burns, Kurihara and Sano apply these methods to the setting of the `Rubin-Stark conjecture', we study the `evaluators' defined in a more general setting by Emmons and Popescu in [7] and by Vallieres in [14]. This allows us to conjecture that the ideals formed from the images of the evaluators can be described precisely in terms of the higher Fitting ideals of the canonical Selmer groups of \(\mathbb{G}_m\). Moreover, we are able to prove that this conjecture follows from the equivariant Tamagawa number conjecture.

Let $(a,b,c)$ be a primitive Pythagorean triple. In 1956, Jeśmanowicz conjectured that the equation $a^{x}+b^{y}=c^{z}$ has the unique solution $(x,y,z)=(2,2,2)$ in positive integers. In 2010 Miyazaki proposed a similar problem. He conjectured that if $(a,b,c)$ is again a primitive Pythagorean triple with $b$ even, then the equation $c^{x}+b^{y}=a^{z}$ with $x$, $y$ and $z$ positive integers has the unique solution $(x,y,z)=(1,1,2)$ if $c=b+1$ and no solutions if $c>b+1$. He also proved that his conjecture is true if $c \equiv 1 \pmod{b}$. We extend Miyazaki's result to the case $c \equiv 1 \pmod{b/2^{\textrm{ord}_{2}(b)}}$.

F. Takens constructed a residual subset of the state space consisting of initial points with historic behaviour for expanding maps on the circle. We prove that this statistical property of expanding maps on the circle is preserved under small random perturbations. The proof is given by establishing a random Markov partition, which follows from a random version of Shub's Theorem on topological conjugacy with the folding maps.

We derive a formula for the signature of the symmetrized Stokes matrix $\mathcal{S}+\mathcal{S}^\mathrm{T}$ for the $tt^*$-Toda equation, reminiscent of a formula of Beukers and Heckmann for the generalized hypergeometric equation. The condition $\mathcal{S}+\mathcal{S}^\mathrm{T} > 0$ is prominent in the work of Cecotti and Vafa on the $tt^*$ equation; using our formula, we show that the Stokes matrices $\mathcal{S}$ satisfying this condition are parameterized by the points of an open convex polytope.

We show a sufficient condition for the defect $\delta (0,f)$ of an analytic function $f(z) = 1 + \sum_{k=1}^{\infty} c_k z^{n_k}$ in the unit disk with Hadamard gaps to vanish. As a consequence, we find that such $f(z)$ whose characteristic function is sufficiently large has no finite defective value.

A trajectory-harps is made of a trajectory for a Kähler magnetic field $\mathbb{B}_{\kappa}$ and an associated variation of geodesics, and a trajectory-horn is made of a geodesic and an associated variation of trajectories. On a Hadamard Kähler manifold $M$ we study thickness and string-angles of trajectory-harps, and study tube-lengths and tube-angles of trajectory-horns. As an application of these we show that two distinct points on the compactification of $M$ with geometric ideal boundary can be joined by a trajectory for $\mathbb{B}_{\kappa}$ if the strength $|\kappa|$ is less than the upper bound of sectional curvatures of $M$.

Let $S$ be a compact complex surface. With the exception of some complex surfaces, it is known that there are only finitely many deformation types of complex surfaces with the same homotopy type as $S$. Let $\mathcal{H}(S)$ be the set of deformation equivalence classes of complex surfaces homotopy equivalent to $S$. We evaluate $\# \mathcal{H}(S)$ when $S$ is a Hopf surface. As a corollary, we construct a sequence of Hopf surfaces $(S_n)$ such that although $\# \mathcal{H}(S_n)$ is finite for all $n$, the sequence $(\# \mathcal{H}(S_n))$ is unbounded.

We investigate a smooth del Pezzo surface of degree 2 which is not Frobenius split. We give a characterization of non-F-split del Pezzo surfaces of degree 2 which exist only if the characteristic of the ground field is 2 or 3. Moreover, we prove that the set of centers of the blow-ups on $\mathbb{P}^{2}$ which gives a non-F-split del Pezzo surface is projectively equivalent to the only complete 7-arc over $\mathbb{F}_{9}$ if the characteristic is 3.

We shall consider some formulas for weighted averages of the generalized Ramanujan sum $\sum_{d|\gcd(k,n)}f(d)g\left({k}/{d}\right)h\left({n}/{d}\right)$ for any arithmetical functions $f,\ g$ and $h$, with the weights concerning completely multiplicative functions, completely additive functions and others.

A family of smooth geometrically irreducible curves violates the Hasse principle if they have local points everywhere, but they possesses no global points. In this paper, we show how to construct non-constant algebraic families of forms of degree $4k$ that violate the Hasse principle. Some examples of non-constant algebraic families of forms of degrees 12 and 24 that violate the Hasse principle are given to illustrate the method.