A 4th-order parabolic equation is researched through the combined application of potential well theory and energy estimation techniques. When $q \gt p-1$, the well-posedness and asymptotic behavior for $J(u_{0}) \lt d$, $I(u_{0}) \gt 0$ or $J(u_{0}) = d$, $I(u_{0}) \geq 0$ and finite-time blow-up phenomena for $J(u_{0}) \lt d$, $I(u_{0}) \lt 0$ or $J(u_{0}) = d$, $I(u_{0}) \lt 0$ are obtained respectively. For $J(u_{0}) \gt d$, based on different initial energy levels, we establish sufficient conditions for the global existence of solutions as well as blow-up phenomena. When $q \leq p-1$, we establish the global existence for $q \lt p-1$ with any $k \gt 0$ or $q = p-1$ with small $k$.

A graph $G$ is said to be crossing-critical if $\operatorname{cr}(G-e) \lt \operatorname{cr}(G)$ for every edge $e$ of $G$, where $\operatorname{cr}(G)$ is the crossing number of $G$. Richter and Thomassen [17] constructed an infinite family of $4$-regular crossing-critical graphs with crossing number $3$. In this article, we present a new infinite family of $4$-regular crossing-critical graphs.

In this paper, we develop two modified hybrid conjugate gradient methods for vector optimization involving three modified conjugate parameters-Fletcher–Reeves (FR) type, Conjugate Descent (CD) type, and Polak–Ribière–Polyak (PRP) type. The modified FR and CD parameters guarantee the sufficient descent condition without special line searches, while the modified PRP offers strong numerical performance. Leveraging these strengths, we propose two modified hybrid conjugate gradient methods that combine the modified FR/CD conjugate parameters with the PRP conjugate parameter. Global convergence results of the two hybrid conjugate gradient methods are established under the strong Wolfe line search strategy without convexity assumptions. Numerical results show the effectiveness and advantages of the proposed methods.

We establish for the first time the explicit curvature formulas for the horizontal and vertical edges of the strong product of two regular graphs. We complement this result with showing that no general formula exists for the curvatures of diagonal edges in terms of the curvatures of the factors, except for a special case. This gives the curvature formulas for all the edges of the strong product of a complete graph and a regular graph. We also present a clear and simple proof of the curvature formulas for all the edges of the Cartesian product of two regular graphs, originally established by Lin, Lu, and Yau [12].

In this paper, we show that weakly symmetric $\tau$-tilting finite algebras have positive definite Cartan matrices, which implies that we can prove $\tau$-tilting infiniteness of weakly symmetric algebras by calculating their Cartan matrices. Similarly, we obtain the sufficient condition on Cartan matrices that selfinjective algebras are $\tau$-tilting infinite. By applying this result, we show that a group algebra of a semidirect product $(\mathbb{Z}/p^{l} \mathbb{Z})^{n} \rtimes H$ is $\tau$-tilting infinite when $p^{l} \geq n$ and $\# \operatorname{IBr}H \geq \min \{p,3\}$, where $p > 0$ is the characteristic of the ground field, $H$ is a subgroup of the symmetric group $\mathfrak{S}_{n}$ of degree $n$, the action of $H$ permutes the entries of $(\mathbb{Z}/p^{l} \mathbb{Z})^{n}$, and $\operatorname{IBr}H$ denotes the set of irreducible $p$-Brauer characters of $H$. Moreover, we show that under the assumption that $p^{l} \geq n$ and $H$ is a $p'$-subgroup of $\mathfrak{S}_{n}$, $\tau$-tilting finiteness of a group algebra of $(\mathbb{Z}/p^{l} \mathbb{Z})^{n} \rtimes H$ is determined by its $p$-hyperfocal subgroup.

We investigate the efficacy of Tikhonov-regularized ensemble Kalman inversion (TEKI) in identifying unknown parameters within partial differential equations (PDEs), under the assumption of a nonlinear mapping from the parameter space to the observation space and with a fixed ensemble size. We conduct a comprehensive analysis of the coupled stochastic differential equations (SDEs) derived from continuous-time formulations, characterizing the ensemble collapse in terms of second moment and almost sure convergence with given rate. Furthermore, this collapse serves as an indicator of the confidence level of our parameter estimator, which is the empirical mean of the particle ensemble.

The main purpose of this paper is to study the pointwise lower dimension of a measure, as a natural dual to the notion of pointwise Assouad dimension introduced by Anttila (2023), by a number of examples. This concept describes the extreme local behavior of a measure around a given point. We verify the exact dimensionality property for quasi-Bernoulli measures supported on self-conformal sets satisfying the strong separation condition. In particular, we compute the pointwise lower dimension explicitly for invariant measures with place-dependent probabilities on self-conformal sets, self-similar doubling measures satisfying the open set condition, and self-affine measures on Bedford–McMullen sponges satisfying the very strong separation condition, which complement the results by Anttila (2023).

For $p \in (1,\alpha) \cup (\alpha,\infty)$, $s \gt 0$, this paper focuses on the continuity of the solution to the weighted $L_{p}$ Minkowski problem for $\alpha$-homogeneous, $s$-concave and even Borel measure with continuous density. We show that a sequence of convex bodies is convergent in Hausdorff metric if the sequence of the weighted $L_{p}$ surface area measures associated with these convex bodies is weakly convergent.

In this paper, we investigate the following nonlinear Biharmonic Choquard equation with Hardy–Littlewood–Sobolev upper critical exponent \[ \begin{cases} \Delta^{2} u = \lambda u + \mu(I_{\alpha} * |u|^{p}) |u|^{p-2} u + (I_{\alpha} * |u|^{4^{*}_{\alpha}}) |u|^{4^{*}_{\alpha}-2} u, &x \in \mathbb{R}^{N}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, dx = \rho, \end{cases} \] where $N \geq 5$, $\mu \gt 0$, $\rho \gt 0$, $0 \lt \alpha \lt N$, $(N+\alpha)/N \lt p \lt 4^{*}_{\alpha} := (N+\alpha)/(N-4)$, $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier and $I_{\alpha}$ is the Riesz potential. Under different assumptions on $p$, we show the existence of normalized solutions by variational method and Pohožaev manifold. Moreover, we describe the behavior of the solutions as $\mu \to 0$.

In the era of big data, real-world data like digital photographs, MRI scans, or speech signals usually has a high dimensionality, we need to reduce the dimensionality adequated in order to save data in the limited storage space and to transmit efficiently. Some common methods of dimensionality reduction such as singular value decomposition, principal component analysis, independent component analysis and autoencoders. However, for high dimensionality data, these methods suffer from heavily computational complexity and memory requirement in practice. Therefore, we provide an efficient and accurate method, which is the numerical solution of the projected nonsymmetric algebraic Riccati equations (pNAREs) arisen in the refinement of estimates of invariant subspaces (REIS). The theoretical contribution of our paper is threefold. Firstly, we introduce how to apply SVD to do image compression and provide some mathematics measurement tools. Secondly, the core of this paper is to adapt REIS and find the first singular values to do image compression, especially for high resolution images. The main idea of REIS is applied to large-scale and real matrices via pNAREs or their associated Sylvester equations through Newton's method. Thirdly, we describe some measurement tools like compression ratio, mean square error, peak signal to noise ratio and structural similarity index to evaluate the performance of the compression factors and the quality of the compressed images. Furthermore, the operation counts and computational complexity are also provided. Finally, we present some numerical experiments about some real world image dataset to show the feasibility of our proposed algorithm.