In this paper, we study the perturbation analysis of a class of composite optimization problems, which is a very convenient and unified framework for developing both theoretical and algorithmic issues of constrained optimization problems. The underlying theme of this paper is very important in both theoretical and computational study of optimization problems. Under some mild assumptions on the objective function, we provide a definition of a strong second order sufficient condition (SSOSC) for the composite optimization problem and also prove that the following conditions are equivalent to each other: the SSOSC and the nondegeneracy condition, the nonsingularity of Clarke's generalized Jacobian of the nonsmooth system at a Karush-Kuhn-Tucker (KKT) point, and the strong regularity of the KKT point. These results provide an important way to characterize the stability of the KKT point. As for the convex composite optimization problem, which is a special case of the general problem, we establish the equivalence between the primal/dual second order sufficient condition and the dual/primal strict Robinson constraint qualification, the equivalence between the primal/dual SSOSC and the dual/primal nondegeneracy condition. Moreover, we prove that the dual nondegeneracy condition and the nonsingularity of Clarke's generalized Jacobian of the subproblem corresponding to the augmented Lagrangian method are also equivalent to each other. These theoretical results lay solid foundation for designing an efficient algorithm.

We examine properties of achievement sets of series in $\mathbb{R}^2$. We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of P-sums as a cut of an achievement set in $\mathbb{R}^2.$ We introduce a notion of the spectre of a set in an Abelian group, which generalizes the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpiński carpet is not an achievement set of any series.

Any finite-dimensional commutative (associative) graded algebra with all nonzero homogeneous subspaces one-dimensional is defined by a symmetric coefficient matrix. This algebraic structure gives a basic kind of $A$-graded algebras originally studied by Arnold. In this paper, we call them diagonally graded commutative algebras (DGCAs) and verify that the isomorphism classes of DGCAs of dimension $\leq 7$ over an arbitrary field are in bijection with the equivalence classes consisting of coefficient matrices with the same distribution of nonzero entries, while dramatically there may be infinitely many isomorphism classes of dimension $n$ corresponding to one equivalence class of coefficient matrices when $n\geq 8$. Furthermore, we adopt the Skjelbred-Sund method of central extensions to study the isomorphism classes of DGCAs, and associate any DGCA with a undirected simple graph to explicitly describe its corresponding second (graded) commutative cohomology group as an affine variety.

We show that there is a PDE formulation in terms of Fokker-Planck equations for weak optimal transport problems. The main novelty is that we introduce a minimization problem involving Fokker-Planck equations in the extended sense of measure-valued solutions and prove that it is equal to the associated weak transport problem.

We use instanton gauge theory to prove that if $Y$ is a closed, orientable $3$-manifold such that $H_1(Y;\mathbb{Z})$ is nontrivial and either $2$-torsion or $3$-torsion, and if $Y$ is neither $\#^r \mathbb{RP}^3$ for some $r\geq 1$ nor $\pm L(3,1)$, then there is an irreducible representation $π_1(Y) \to \mathrm{SL}(2,\mathbb{C})$. We apply this to show that the Kauffman bracket skein module of a non-prime 3-manifold has nontrivial torsion whenever two of the prime summands are different from $\mathbb{RP}^3$, answering a conjecture of Przytycki (Kirby problem 1.92(F)) unless every summand but one is $\mathbb{RP}^3$. As part of the proof in the $2$-torsion case, we also show that if $M$ is a compact, orientable $3$-manifold with torus boundary whose rational longitude has order 2 in $H_1(M)$, then $M$ admits a degree-1 map onto the twisted $I$-bundle over the Klein bottle.

The quality and quantity of data used for training greatly influence the performance and effectiveness of deep learning models. In the context of error correction, it is essential to generate high-quality samples that are neither excessively noisy nor entirely correct but close to the decoding region's decision boundary. To accomplish this objective, this paper utilizes a restricted version of a recent result on Importance Sampling (IS) distribution for fast performance evaluation of linear codes. The IS distribution is used over the segmented observation space and integrated with active learning. This combination allows for the iterative generation of samples from the shells whose acquisition functions, defined as the error probabilities conditioned on each shell, fall within a specific range. By intelligently sampling based on the proposed IS distribution, significant improvements are demonstrated in the performance of BCH(63,36) and BCH(63,45) codes with cycle-reduced parity-check matrices. The proposed IS-based-active Weight Belief Propagation (WBP) decoder shows improvements of up to 0.4dB in the waterfall region and up to 1.9dB in the error-floor region of the BER curve, over the conventional WBP. This approach can be easily adapted to generate efficient samples to train any other deep learning-based decoder.

In this paper, we give a characterization of Coxeter group representations of Lusztig's a-function value 1, then determine all the irreducible such representations for certain simply laced Coxeter groups.

Let C(a) be the central Cantor set generated by a sequence a with terms in (0,1). It is known that the difference set C(a)-C(a) of C(a) can has one of three possible forms: a finite union of closed intervals, a Cantor set, or a Cantorval. In the previous paper there was proved a sufficient condition for the sequence a which implies that C(a) - C(a) is a Cantorval. In this paper we give different conditions for a sequence a which guarantee the same assertion. We also prove a corollary, which provides infinitely many new examples of Cantorvals.

We revisit the approach to the lower critical dimension $d_{\rm lc}$ in the Ising-like $φ^4$ theory within the functional renormalization group by studying the lowest approximation levels in the derivative expansion of the effective average action. Our goal is to assess how the latter, which provides a generic approximation scheme valid across dimensions and found to be accurate in $d\geq 2$, is able to capture the long-distance physics associated with the expected proliferation of localized excitations near $d_{\rm lc}$. We show that the convergence of the fixed-point effective potential is nonuniform when $d\to d_{\rm lc}$ with the emergence of a boundary layer around the minimum of the potential. This allows us to make analytical predictions for the value of the lower critical dimension $d_{\rm lc}$ and for the behavior of the critical temperature as $d\to d_{\rm lc}$, which are both found in fair agreement with the known results. This confirms the versatility of the theoretical approach.

We explore the possibility that an $SU(2)_L$ triplet scalar with hypercharge $Y=0$ is the origin of the $95\,$GeV diphoton excess. For a small mixing angle with the Standard Model Higgs, its neutral component has naturally a sizable branching ratio to $γγ$ such that its Drell-Yan production via $pp\to W^*\to H H^\pm$ is sufficient to obtain the desired signal strength, where $H^\pm$ is the charged Higgs component of the triplet. The predictions of this setup are: 1) The $γγ$ signal has a $p_T$ spectrum different from gluon fusion but similar to associated production. 2) Photons are produced in association with tau leptons and jets, but generally do not fall into the vector-boson fusion category. 3) The existence of a charged Higgs with $m_{H^\pm}\approx\!(95\pm5)\,$GeV leading to $σ(pp\to ττνν)\approx0.4\,$pb, which is of the same level as the current limit and can be discovered with Run 3 data. 4) A positive definite shift in the $W$ mass as suggested by the current global electroweak fit.

We study the frequency non-reciprocity of the spin waves in symmetric CoFeB/Ru/CoFeB synthetic antiferromagnets stacks set in the scissors state by in-plane applied fields. Using a combination of Brillouin Light Scattering and propagating spin wave spectroscopy experiments, we show that the acoustical spin waves in synthetic antiferromagnets possess a unique feature if their wavevector is parallel to the applied field: the frequency non-reciprocity due to layer-to-layer dipolar interactions can be so large that the acoustical spin waves transfer energy in a unidirectional manner for a wide and bipolar interval of wavevectors. Analytical modeling and full micromagnetic calculations are conducted to account for the dispersion relations of the optical and acoustical spin waves for arbitrary field orientations. Our formalism provides a simple and direct method to understand and design devices harnessing unidirectional propagation of spin waves in synthetic antiferromagnets.

In this paper, we extend some classical results of the Szego theory of orthogonal polynomials on the unit circle to the infinite-dimensional case, and we establish the corresponding Szego limit theorem.

Let $\cC$ be a set of curves in the plane such that no three curves in $\cC$ intersect at a single point and every pair of curves in $\cC$ intersect at exactly one point which is either a crossing or a touching point. János Pach conjectured that the number of pairs of curves in $\cC$ that touch each other is $O(|\cC|)$. We prove this conjecture for $x$-monotone curves.

Despite a great deal of effort in searching for the triplet-like Higgses in the type-II seesaw model, evidence for their production is yet to be found at the LHC. As such, one might be in the balance regarding this model's relevance at the electroweak scale. In this work, we peruse a scenario, akin to compressed mass spectra in Supersymmetry, which might have eluded the experimental searches thus far. We perform a multivariate analysis to distinguish signals with a pair of same-sign leptons with low invariant mass from the SM processes, including those accruing from fake leptons and electron charge misidentification, and find that a significant part of the hitherto unconstrained parameter space could be probed with the already collected Run 2 13 TeV LHC and future HL-LHC data.

How nuclides decay in the superheavy region is key information for investigating new elements beyond oganesson and the island of stability. The Random Forest algorithm is applied to study the competition between different decay modes in the superheavy region, including $α$ decay, $β^-$ decay, $β^+$ decay, electron capture and spontaneous fission. The observed half-lives and dominant decay mode are well reproduced. The dominant decay mode of 96.9 % nuclei beyond $^{212}$Po is correctly described. $α$ decay is predicted to be the dominant decay mode for isotopes in new elements $Z = 119 - 122$, except for spontaneous fission in some even-even ones because of the odd-even staggering effect. The predicted half-lives show the existence of a long-lived spontaneous fission island at the southwest of $^{298}$Fl caused by the competition of nuclear deformation and Coulomb repulsion. More understanding of spontaneous fission, especially beyond $^{286}$Fl, is crucial to search for new elements and the island of stability.

We prove a nonuniqueness theorem for Bernoulli site percolation on properly embedded planar graphs, and we obtain a general connectivity principle beyond planarity. Let $G$ be an infinite connected graph properly embedded in $\RR^2$ with minimum degree at least $7$. Then \[ p_c^{\mathrm{site}}(G)<\tfrac12, \] and for every \[ p\in \bigl(p_c^{\mathrm{site}}(G),\,1-p_c^{\mathrm{site}}(G)\bigr), \] Bernoulli$(p)$ site percolation on $G$ has almost surely infinitely many infinite open clusters. In particular, this verifies a conjecture of Benjamini and Schramm for properly embedded planar graphs. The core new ingredient is an explicit embedded-tree separation mechanism for planar nonuniqueness. We construct embedded trees and an embedded forest whose separation properties yield exponential decay of two-point connection probabilities in the matching graph. To treat the high-density regime, we introduce a binary-tree version of uniform percolation and prove stability of infinite clusters under edge additions, without any bounded-degree assumption. Beyond the planar theorem, we prove a general lower bound on two-point connectivity under uniqueness for arbitrary infinite locally finite graphs. As a consequence, if \[ p_c^{\mathrm{site}}(G)<p<p_{\mathrm{conn}}(G), \] then Bernoulli site percolation on $G$ has almost surely infinitely many infinite open clusters.

We study price discovery in a model where an informed agent has arbitrary private information about state probabilities and trades state-contingent claims. The model unifies the key elements of Arrow-Debreu (1954) and Kyle (1985). When the claims are options, the informed agent has arbitrary information about the underlying asset's payoff distribution and trades option portfolios. Our setting provides the first equilibrium framework that encompasses longs-tanding option-market practices and regularities, including common trading strategies and the volatility smile across strikes.

We investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \mathbb{N}$, $l+r<p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the form: the interval $[-1,1]$, a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval. As corollaries we give examples of Cantor sets and Cantorvals, which can be easily described using some positional numeral systems.

The weighted Euclidean norm $\|x\|_w$ of a vector $x\in \mathbb{R}^d$ with weights $w\in \mathbb{R}^d$ is the Euclidean norm where the contribution of each dimension is scaled by a given weight. Approaches to dimensionality reduction that satisfy the Johnson-Lindenstrauss (JL) lemma can be easily adapted to the weighted Euclidean distance if weights are known and fixed: it suffices to scale each dimension of the input vectors according to the weights, and then apply any standard approach. However, this is not the case when weights are unknown during the dimensionality reduction or might dynamically change. In this paper, we address this issue by providing a linear function that maps vectors into a smaller complex vector space and allows to retrieve a JL-like estimate for the weighted Euclidean distance once weights are revealed. Our results are based on the decomposition of the complex dimensionality reduction into several Rademacher chaos random variables, which are studied using novel concentration inequalities for sums of independent Rademacher chaoses.

In this work, we propose an efficient implementation of mixtures of experts distributional regression models which exploits robust estimation by using stochastic first-order optimization techniques with adaptive learning rate schedulers. We take advantage of the flexibility and scalability of neural network software and implement the proposed framework in mixdistreg, an R software package that allows for the definition of mixtures of many different families, estimation in high-dimensional and large sample size settings and robust optimization based on TensorFlow. Numerical experiments with simulated and real-world data applications show that optimization is as reliable as estimation via classical approaches in many different settings and that results may be obtained for complicated scenarios where classical approaches consistently fail.

Search for light (within the mass range 84-200 GeV) doubly-charged Higgs bosons decaying into a pair of W-bosons has been deemed challenging using the conventional LHC searches with leptons, jets and missing transverse momentum in the final state. Such Higgses together with slightly heavier singly-charged and neutral Higgses, when arranged in an $SU(2)_L$ triplet as in the type-II see-saw model, are lately shown to accommodate the recent measurement of the $W$-boson mass by the CDF collaboration. These, when produced in a highly Lorentz-boosted regime, tend to manifest themselves as a single fat-jet or a pair of adjacent same-sign leptons plus missing transverse momentum. First, we perform a multivariate analysis to discern such exotic jets from the SM jets. Then, we present a novel search in the final state with an exotic jet and two same-sign leptons plus missing transverse momentum. We find that such low-mass doubly-charged Higgsses could be directly probed with the already collected Run 2 LHC data.

We introduce a rigorous and sensitive significance test for hyperuniformity that yields reliable results even from a single sample. Our approach is based on a detailed analysis of the empirical Fourier transform of a stationary point process in $\mathbb{R}^d$. For large system sizes, we derive the asymptotic covariances and establish a multivariate central limit theorem (CLT) for these empirical Fourier transforms. Their absolute square value, the scattering intensity, is then used as the standard estimator of the structure factor. The above CLT holds for a preferably large class of point processes, and whenever this is the case, the scattering intensity satisfies a multivariate limit theorem as well. Hence, we can use the likelihood ratio principle to test for hyperuniformity. Remarkably, the asymptotic distribution of the resulting test statistic is universal under the null hypothesis of hyperuniformity. We obtain its explicit form from simulations with very high accuracy. The novel test precisely keeps a nominal significance level for hyperuniform models, and it rejects non-hyperuniform examples with high power even in borderline cases. Moreover, it does so given only a single sample with a practically relevant system size.

For a post-critically finite hyperbolic rational map $f$, we show that its Julia set $\mathcal{J}_f$ has Ahlfors-regular conformal dimension one if and only if $f$ is a crochet map, i.e., there is an $f$-invariant connected graph $G$ containing the post-critical set such that $f|_G$ has topological entropy zero. We use finite subdivision rules to obtain graph virtual endomorphisms, which are 1-dimensional models of post-critically finite rational maps, and we approximate the asymptotic conformal energies of graph virtual endomorphisms to estimate the Ahlfors-regular conformal dimensions of Julia sets. To prove the main theorem, we also establish the monotonicity of asymptotic conformal energies under the decomposition of rational maps by invariant multicurves.

For $1<p\le 2$, we establish sharp inequalities for the Fourier transform of the characteristic function of the $l^p$-unit ball $B_p\subset\mathbb{R}^2$. We show that $$ \sup_{\boldsymbolω \in \mathbb{R}^2} \|\boldsymbolω \|_2^{3/2}|\widehat{χ_{B_p}} (\boldsymbolω)| \asymp (p-1)^{-1/2} \quad \text{as } p\rightarrow1+ $$ As an application, we obtain corresponding bounds for lattice point discrepancy inequalities for dilates of $B_p$.

We consider the subspaces $c$, $\widehat{c}$, $S$ of $\ell^\infty$, where $\widehat{c}$ consists of almost convergent sequences, and $S$ consists of sequences whose arithmetic means of consecutive terms are convergent. We know that $c\subset \widehat{c} \subset S$. We examine the largeness of $c$ in $\widehat{c}$, $\widehat{c}$ in $S$ and $S$ in $\ell^\infty$. We will do it from the viewpoints of porosity, algebrability and measure.

The width for the mu decay is calculated in the V-A theory leaving open the possibility of non zero neutrino masses. It is shown that not only the agreement with the experimental data is kept, but the smallness of the experimental error allows to improve the constraint of nu mass (muon based) down to 0.021 MeV, provided that nu mass (electron based) is as low as indicated by the 3H beta decay. An analogous constraint for the nu mass (tau based) is not possible since in this case the decay width has a larger experimental error.

The calculation of the $μ$ decay is performed in V-A theory taking into account that, in agreement with the observation of the oscillations, the neutrinos must be massive. Provided that the masses of the electron and muon neutrinos are not higher than the known upper values, the result is a possible modification of the Fermi constant, no effect at all on the polarization of the outgoing electron and possibly an endpoint effect on the electron energy and angle distributions.

It is a challenge to reproduce the full 6D space-phase properties of Sagittarius (Sgr) dwarf galaxy and its Stream simultaneously. Using N-body simulations with a Milky Way mass of 5.2$\times10^{11}$ M$_{\odot}$ and a ``scaling down'' Sgr mass of 9.3$\times10^{8}$ M$_{\odot}$, from a qualitative point of view, we have been able to reproduce well all 3D spatial features of Sgr stream, including its core, leading and trailing arms, and their associated bifurcations, moreover, the overall trend of the reported 3D kinematics properties of the Sgr stream have also been reproduced without fine tuning. Furthermore, we also find that our model fails in reproducing the exact behaviours of the line-of-sight velocity and angular-energy distributions. It let us to suggest that significant further progress might be achievable after introducing a major component in the Sgr progenitor, which is the gas that dominates all Irregular dwarf galaxies in the Sgr mass range and can slow down the radial velocity of Sgr before its removal, if gas can not solve this problem then we will consider a non-spherical Milky Way halo with hot gas, LMC, etc. As the first step for us to towards the complete understanding of the Sgr system, this progress is also advancing our understanding of the bifurcations, the generation of which might be due to the MW shocks at each pericenter passage, and also be linked to the orientation and disk-shape in the initial conditions.

Exotic leptons in large gauge multiplets, appearing in many scenarios beyond the Standard Model (SM), can be produced at the LHC in pairs or association. Owing to their large masses, their eventual decay products -- SM leptons and bosons -- tend to be highly boosted, with the jets stemming from the SM bosons more likely to manifest themselves as a single fat-jet rather than two resolved ones. With the corresponding SM backgrounds being suppressed, final states with two or three leptons and one or two fat-jets are expected to be sensitive in probing exotic fermions much heavier than 1 TeV, and we propose and investigate an appropriate search strategy. To concentrate on the essential, we consider extensions of the SM by leptonic multiplets of a single kind (triplets, quadruplets or quintuplets), bearing in mind that such simplified models typically arise as low-energy limits of more ambitious scenarios addressing various lacunae of the SM. Performing a systematic and comprehensive study of nine such scenarios at the 13 TeV LHC, we find that the corresponding $5σ$ discovery reaches a range from 985 GeV to 1650 GeV (1345 GeV to 2020 GeV) for 300 (3000) fb$^{-1}$.

In the paper we apply some of the results from the theory of ball spaces in the semimetric spaces. This allowed us to obtain some fixed point theorems which we believe to be unknown to this day. We also show the limitations of the ball space approach to this topic. As a byproduct, we obtain the equivalence of some different notions of completness in semimetric spaces where the distance function is $1$-continuous. In the second part of the article, we generalize Caristi-Kirk results for for $b$-metric spaces. Additionally, we obtain characterization of semicompleteness for $1$-continuous $b$-metric space via fixed point theorem analogous to the result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.