arXiv Math @arxiv_math@qoto.org

Mobile edge generation (MEG) is an emerging technology that allows the network to meet the challenging traffic load expectations posed by the rise of generative artificial intelligence~(GAI). A novel MEG model is proposed for deploying GAI models on edge servers (ES) and user equipment~(UE) to jointly complete text-to-image generation tasks. In the generation task, the ES and UE will cooperatively generate the image according to the text prompt given by the user. To enable the MEG, a pre-trained latent diffusion model (LDM) is invoked to generate the latent feature, and an edge-inferencing MEG protocol is employed for data transmission exchange between the ES and the UE. A compression coding technique is proposed for compressing the latent features to produce seeds. Based on the above seed-enabled MEG model, an image quality optimization problem with transmit power constraint is formulated. The transmitting power of the seed is dynamically optimized by a deep reinforcement learning agent over the fading channel. The proposed MEG enabled text-to-image generation system is evaluated in terms of image quality and transmission overhead. The numerical results indicate that, compared to the conventional centralized generation-and-downloading scheme, the symbol number of the transmission of MEG is materially reduced. In addition, the proposed compression coding approach can improve the quality of generated images under low signal-to-noise ratio (SNR) conditions.

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for biharmonic equations with built-in stabilizers. Unlike existing stabilizer-free WG methods limited to convex elements in finite element partitions, our approach accommodates both convex and non-convex polytopal meshes, offering enhanced versatility. It employs bubble functions without the restrictive conditions required by existing stabilizer-free WG methods, thereby simplifying implementation and broadening application to various partial differential equations (PDEs). Additionally, our method supports flexible polynomial degrees in discretization and is applicable in any dimension, unlike existing stabilizer-free WG methods that are confined to specific polynomial degree combinations and 2D or 3D settings. We demonstrate optimal order error estimates for WG approximations in both a discrete $H^2$ norm for $k\geq 2$ and a $L^2$ norm for $k>2$, as well as a sub-optimal error estimate in $L^2$ when $k=2$, where $k\geq 2$ denotes the degree of polynomials in the approximation.

We present Russell's antinomy using three distinct deductive systems, which are then compared to deepen the logical deductions that lead to the contradiction. Some inferential paths are then presented, alternative to the commonly accepted one, that allow for the formal assertion of the antinomy without deriving the contradiction, thus preserving the coherence of the system. In light of this, the purpose of this article is to propose a review of the consequences of asserting Russell's antinomy and, by extension, the widespread belief that any attempt to resolve a paradox is doomed to failure.

We study the (generalized) one-dimensional population model developed by Anguige \& Schmeiser [1], which reflects cell-cell adhesion and volume filling under no-flux boundary condition. In this generalized model, depending on the adhesion and volume filling parameters $α,β\in[0,1],$ the resulting equation is classified into six types. Among these, we focus on the type exhibiting strong effects of both adhesion and volume filling, which results in a class of advection-diffusion equations of the forward-backward-forward type. For five distinct cases of initial maximum, minimum and average population densities, we derive the corresponding patterns for the global behavior of weak solutions to the initial and no-flux boundary value problem. Due to the presence of a negative diffusion regime, we indeed prove that the problem is ill-posed and admits infinitely many global-in-time weak solutions, with the exception of one specific case of the initial datum. This nonuniqueness is inherent in the method of convex integration that we use to solve the Dirichlet problem of a partial differential inclusion arising from the ill-posed problem.

In this short article, we will reconstruct the KP equation from Plucker relations and provide some generalizations on this topic. Additionally, in the final section, we define the discrete function $τ$ in a similar manner, leading to the construction of an integer sequence that has not yet been listed in the OEIS. Furthermore, this approach allows us to construct many other sequences that are not listed in the OEIS.

This addendum presents a relevant stronger consequence of the main theorem of the paper "Higher order stroboscopic averaged functions: a general relationship with Melnikov functions" [arXiv:2011.03663], EJQTDE No. 77 (2021).

A matching $M$ in a graph $G$ is {\em connected} if $G$ has an edge linking each pair of edges in $M$. The problem to find large connected matchings in graphs $G$ with $α(G)=2$ is closely related to Hadwiger's conjecture for graphs with independence number 2. The problem of finding a large connected matching in a general graph is NP-hard. F{ü}redi et al. in 2005 conjectured that each $(4t-1)$-vertex graph $G$ with $α(G)=2$ contains a connected matching of size at least $t$. Cambie recently showed that if this conjecture is false, then so is Hadwiger's conjecture. In this paper, we present a number of properties possessed by a counterexample to F{ü}redi et al.'s conjecture, and then using these properties, we prove that F{ü}redi et al.'s conjecture holds for $t\leq22$.

A graph $G$ is said to be Ramsey size-linear if $r(G,H) =O_G (e(H))$ for every graph $H$ with no isolated vertices. Erdős, Faudree, Rousseau, and Schelp observed that $K_4$ is not Ramsey size-linear, but each of its proper subgraphs is, and they asked whether there exist infinitely many such graphs. In this short note, we answer this question in the affirmative.

We study the property of uniform discreteness within discrete orbits of non-uniform lattices in $SL_2(\mathbb{R})$, acting on $\mathbb{R}^2$ by linear transformations. We provide a new proof of the conditions under which the orbit of a non-uniform lattice in $SL_2(\mathbb{R})$ is uniformly discrete, by using Diophantine properties. Our results include a detailed analysis of the asymptotic behavior of the error terms. Focusing on a specific group $Γ$ and a discrete orbit of it, $S$, the main result of this paper is that for any $ε>0$, three points in $S$ can be found on a horizontal line within distance $ε$ of each other. This gives a partial result toward a conjecture of Lelièvre. The set $S$ and group $Γ$ are respectively the set of long cylinder holonomy vectors, and Veech group, of the "golden L" translation surface.

We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n))_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb Z[x]$ that is a product of distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant $κ_P>0$ such that \[ \frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu. We also study large fluctuations of $\sum_{n\leq N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that \[ \Big|\sum_{n\leq N}f(n^2+1)\Big|\gg \sqrt{N \log\log N}. \] This matches the bound one expects from the law of iterated logarithm.

Let $\mathcal{C}$ be a small category and let $R$ be a dg-representation of the category $\mathcal{C}$, that is, a pseudofunctor from a small category to the category of small dg $k$-categories, where $k$ is a commutative unital ring. In this paper, we mainly study the category $\mbox{Mod-} R$ of right modules over $R$. We characterize it as an ordinary category of dg-modules over a (differential graded) dg-category $Gr(R)$, where $Gr(R)$ is the linear Grothendieck construction of $R$. This characterization generalizes the Theorem 3.18 of the paper (S. Estrada and S. Virili. Cartesian modules over representations of small categories. Adv. in Math. 310: 557-609, 2017) of Estrada and Virili to the dg-category context. Furthermore, as some applications of the main characterization theorem, we classify the hereditary torsion pairs, (split) TTF triples and Abelian recollements in $\mbox{Mod-} R$ respectively.

The Wigner caustic and the Centre Symmetry Set of a closed smooth planar curve are known singular sets which generically admit only cusp singularities. Applications of these objects in semi-classical quantum physics, in chaos theory, in singularity theory, in convex geometry, have been studied since the 1970s until today. These sets can be viewed as envelopes of special families of lines and thanks to that they have many geometric artistic values.

We show that the Carathéodory number of the joint numerical range of $d$ many bounded self-adjoint operators is at most $d-1$, and even at most $d-2$ if the underlying Hilbert space has dimension at least $3$. This extension of the classical convexity results for numerical ranges shows that also joint numerical ranges are significantly less non-convex than general sets.

Harmonic synthesis describes translation invariant linear spaces of continuous complex valued functions on locally compact abelian groups. The basic result due to L. Schwartz states that such spaces on the reals are topologically generated by the exponential monomials in the space -- in other words the locally compact abelian group of the reals is synthesizable. This result does not hold for continuous functions in several real variables as it was shown by D.I. Gurevich's counterexamples. On the other hand, if two discrete abelian groups have this synthesizability property, then so does their direct sum, as well. In this paper we show that if two locally compact abelian groups have this synthesizability property and at least one of them is discrete, then their direct sum is synthesizable. In fact, more generally, we show that any extension of a synthesizable locally compact abelian group by a synthesizable discrete abelian group is synthesizable. This is an important step toward the complete characterisation of synthesizable locally compact abelian groups.

Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}m^2$ and it has atleast six distinct prime divisors. Also we prove that $N$ must be atleast $2\cdot 10^{12}$. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-5$ and if $Ω(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $Ω(n)$ and $ω(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$.

K. S. Vorushilov described Jordan-Kronecker invariants for semi-direct sums $\operatorname{sl} \ltimes \left(\mathbb{C}^n\right)^k$ if $k > n$ or if $n$ is a multiple of $k$. We describe the Jordan-Kronecker invariants in the cases $n \equiv \pm 1 \pmod k$.

As a popular form of knowledge and experience, patterns and their identification have been critical tasks in most data mining applications. However, as far as we are aware, no study has systematically examined the dynamics of pattern values and their reuse under varying conditions. We argue that when problem conditions such as the distributions of random variables change, the patterns that performed well in previous circumstances may become less effective and adoption of these patterns would result in sub-optimal solutions. In response, we make a connection between data mining and the duality theory in operations research and propose a novel scheme to efficiently identify patterns and dynamically quantify their values for each specific condition. Our method quantifies the value of patterns based on their ability to satisfy stochastic constraints and their effects on the objective value, allowing high-quality patterns and their combinations to be detected. We use the online bin packing problem to evaluate the effectiveness of the proposed scheme and illustrate the online packing procedure with the guidance of patterns that address the inherent uncertainty of the problem. Results show that the proposed algorithm significantly outperforms the state-of-the-art methods. We also analysed in detail the distinctive features of the proposed methods that lead to performance improvement and the special cases where our method can be further improved.

In this paper, we proved that for every non-degenerate $C^3$ compact star-shaped hypersurface $Σ$ in $\mathbb{R}^{8}$ which carries no prime closed characteristic of Maslov-type index $-1$, there exist at least four prime closed characteristics on $Σ$.

The ranking of possible alternatives during the design or operation of an industrial process leads to the definition of a solution representing the best compromise between several contradictory objectives. The choice of solution is a decision that can be modeled using knowledge of the corresponding preferences. The decision and preference model used is that of flow balances. Munster cheese is used as an example. These are highly distinctive cheeses that are not suitable for uninformed consumers. If the decision-maker wishes to widen his consumer panel, he must modify the production line settings. In the event of a change in preferences due to an external influence, the decision-maker must adapt, but not too abruptly, to avoid both production line oscillations and consumer annoyance. To solve this problem, we propose a new technique: dynamic flow balances. This makes it possible to forecast production trends. Prospects are also proposed, such as the creation of a multi-criteria predictive control.

Torque-driven microscale swimming robots, or microrotors, hold significant potential in biomedical applications such as targeted drug delivery, minimally invasive surgery, and micromanipulation. This paper addresses the challenge of controlling the transport of fluid volumes using the flow fields generated by interacting groups of microrotors. Our approach uses polynomial chaos expansions to model the time evolution of fluid particle distributions and formulate an optimal control problem, which we solve numerically. We implement this framework in simulation to achieve the controlled transport of an initial fluid particle distribution to a target destination while minimizing undesirable effects such as stretching and mixing. We consider the case where translational velocities of the rotors are directly controlled, as well as the case where only torques are controlled and the rotors move in response to the collective flow fields they generate. We analyze the solution of this optimal control problem by computing the Lagrangian coherent structures of the associated flow field, which reveal the formation of transport barriers that efficiently guide particles toward their target. This analysis provides insights into the underlying mechanisms of controlled transport.