arXiv Math @arxiv_math@qoto.org

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.

This paper introduces an innovative approach for signal reconstruction using data acquired through multi-input-multi-output (MIMO) sampling. First, we show that it is possible to perfectly reconstruct a set of periodic band-limited signals $\{x_r(t)\}_{r=1}^R$ from the samples of $\{y_m(t)\}_{m=1}^M$, which are the output signals of a MIMO system with inputs $\{x_r(t)\}_{r=1}^R$. Moreover, an FFT-based algorithm is designed to perform the reconstruction efficiently. It is demonstrated that this algorithm encompasses FFT interpolation and multi-channel interpolation as special cases. Then, we investigate the consistency property and the aliasing error of the proposed sampling and reconstruction framework to evaluate its effectiveness in reconstructing non-band-limited signals. The analytical expression for the averaged mean square error (MSE) caused by aliasing is presented. Finally, the theoretical results are validated by numerical simulations, and the performance of the proposed reconstruction method in the presence of noise is also examined.

In this paper, we construct a $p$-adic $L$-function for a $P$-ordinary Hida family of cuspidal automorphic representations on a unitary group. We first describe the geometry and representation theory involved in the study of such $P$-ordinary automorphic representations. Then, we construct a $p$-adic family of Siegel Eisenstein series and use the doubling method of Garrett and Piatetski--Shapiro-Rallis to relate certain zeta integrals to special values of standard $L$-functions. The $p$-adic interpolation of these special values is obtained by looking at the $p$-adic variation of the Fourier coefficients of the Siegel Eisenstein series.

The permutations of horse racing, where ties are possible, are counted by the $\textit{Fubini numbers}$, also called the $\textit{horse numbers}$. The $\textit{r-horse numbers}$ are a counting of such horse race finishes where some subset of $r$ horses agree to finish the race in a specific strong ordering. The $r$-horse numbers for fixed $r$ are expressed as a sum of $r$ index shifted sequences of Fubini numbers weighted with the $\textit{signed Stirling numbers of the first kind}$. Then eventual modular periodicity of $\textit{r-Fubini numbers}$ is shown and their maximum period is determined to be the $\textit{Carmichael function}$ of the modulus. The maximum period is attained in the case of an odd modulus for Fubini numbers.

In this paper, we calculate the annihilating polynomial of the colored Jones polynomial for the Hopf link and the Whitehead link and consider the link version of the AJ conjecture. We also give another annihilating polynomial of the colored Jones polynomial for those links.

In this article, we develop and present a novel regularization scheme for ill-posed inverse problems governed by nonlinear partial differential equations (PDEs). In [43], the author suggested a bi-level regularization framework. This study significantly improves upon the bi-level algorithm by sequential initialization, yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect with respect to noise and allows for the usage of inexact PDE solvers. The sequential bi-level approach illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined. To demonstrate the applicability of our algorithms, we moreover prove that the proposed tangential cone condition is satisfied.

This paper studies the interference broadcast channel comprising multiple multi-antenna Base Stations (BSs), each controlling a beyond diagonal Reconfigurable Intelligent Surface (RIS) and serving multiple single-antenna users. Wideband transmissions are considered with the objective to jointly design the BS linear precoding vectors and the phase configurations at the RISs in a distributed manner. We take into account the frequency selectivity behavior of each RIS's tunable meta-element, and focusing on the sum rate as the system's performance criterion, we present a distributed optimization approach that enables cooperation between the RIS control units and their respective BSs. According to the proposed scheme, each design variable can be efficiently obtained in an iterative parallel way with guaranteed convergence properties. Our simulation results demonstrate the validity of the presented distributed algorithm and showcase its superiority over a non-cooperative scheme as well as over the special case where the RISs have a conventional diagonal structure.

We investigate the structure of the minimal displacement set in weakly systolic complexes. We show that such set is systolic and that it embeds isometrically into the complex. As corollaries, we prove that any isometry of a weakly systolic complex either fixes the barycentre of some simplex (elliptic case) or it stabilizes a thick geodesic (hyperbolic case).

In the analysis of parametrized nonautonomous evolutionary equations, bounded entire solutions are natural candidates for bifurcating objects. Appropriate explicit and sufficient conditions for such branchings, however, require to combine contemporary functional analytical methods from the abstract bifurcation theory for Fredholm operators with tools originating in dynamical systems. This paper establishes alternatives classifying the shape of global bifurcating branches of bounded entire solutions to Carathéodory differential equations. Our approach is based on the parity associated to a path of index 0 Fredholm operators, the global Evans function as a recent tool in nonautonomous bifurcation theory and suitable topologies on spaces of Carathéodory functions.

We compute the equivariant K-theory of torus fixed points of Cherkis bow varieties of affine type A. We deduce formulas for the generating series of the Euler numbers of these varieties and observe their modularity in certain cases. We also obtain refined formulas on the motivic level for a class of bow varieties strictly containing Nakajima quiver varieties. These series hence generalise results of Nakajima-Yoshioka. As a special case, we obtain formulas for certain Zastava spaces. We define a parabolic analogue of Nekrasov's partition function and find an equation relating it to the classical partition function.

In this paper, we investigate the subelliptic nonlocal Brezis-Nirenberg problem on stratified Lie groups involving critical nonlinearities, namely, \begin{align*} (-Δ_{\mathbb{G}, p})^s u&= μ|u|^{p_s^*-2}u+λh(x, u) \quad \text{in}\quad Ω, \\ u&=0\quad \text{in}\quad \mathbb{G}\backslash Ω, \end{align*} where $(-Δ_{\mathbb{G}, p})^s$ is the fractional $p$-sub-Laplacian on a stratified Lie group $\mathbb{G}$ with homogeneous dimension $Q,$ $Ω$ is an open bounded subset of $\mathbb{G},$ $s \in (0,1)$, $\frac{Q}{s}>p\geq2,$ $p_s^*:=\frac{pQ}{Q-ps}$ is subelliptic fractional Sobolev critical exponent, $μ, λ>0$ are real parameters and $h$ is a lower order perturbation of the critical power $|u|^{p_s^*-2}u$. Utilising direct methods of the calculus of variation, we establish the existence of at least one weak solution for the above problem under the condition that the real parameter $λ$ is sufficiently small. Additionally, we examine the problem for $μ= 0$, representing subelliptic nonlocal equations on stratified Lie groups depending on one real positive parameter and involving a subcritical nonlinearity. We demonstrate the existence of at least one solution in this scenario as well. We emphasize that the results obtained here are also novel for $p=2$ even for the Heisenberg group.