Main Theorem Research Articles

Hamiltonian facets of classical gauge theories on E-manifolds

Manifolds with boundary, with corners, b-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds were introduced in Nest and Tsygan (2001 Asian J. Math. 5 599–635) and investigated in depth in Miranda and Scott (2021 Rev. Mat. Iberoam. 37 1207–24). In this article we explore their physical facets by extending gauge theories to the E-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E-symplectic manifold. Following the scheme inaugurated in Weinstein (1978 Lett. Math. Phys. 2 417–20), we show the existence of a universal model for a particle interacting with an E-gauge field. In addition, we generalise the description of phase spaces in Yang–Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in Montgomery (1986 PhD Thesis University of California, Berkeley), for base manifolds endowed with an E-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong’s equations, which describe the interaction of a particle with a Yang–Mills field, become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole.

Faster Property Testers in a Variation of the Bounded Degree Model

Property testing algorithms are highly efficient algorithms that come with probabilistic accuracy guarantees. For a property P , the goal is to distinguish inputs that have P from those that are far from having P with high probability correctly, by querying only a small number of local parts of the input. In property testing on graphs, the distance is measured by the number of edge modifications (additions or deletions) that are necessary to transform a graph into one with property P . Much research has focused on the query complexity of such algorithms, i. e., the number of queries the algorithm makes to the input, but in view of applications, the running time of the algorithm is equally relevant. In (Adler, Harwath, STACS 2018), a natural extension of the bounded degree graph model of property testing to relational databases of bounded degree was introduced, and it was shown that on databases of bounded degree and bounded tree-width, every property that is expressible in monadic second-order logic with counting (CMSO) is testable with constant query complexity and sublinear running time. It remains open whether this can be improved to constant running time. In this article we introduce a new model, which is based on the bounded degree model, but the distance measure allows both edge (tuple) modifications and vertex (element) modifications. We show that every property that is testable in the classical model is testable in our model with the same query complexity and running time, but the converse is not true. Our main theorem shows that on databases of bounded degree and bounded tree-width, every property that is expressible in CMSO is testable with constant query complexity and constant running time in the new model. Our proof methods include the semilinearity of the neighborhood histograms of databases having the property and a result by Alon (Proposition 19.10 in Lovász, Large networks and graph limits, 2012) that states that for every bounded degree graph \(\mathcal {G}\) there exists a constant size graph \(\mathcal {H}\) that has a similar neighborhood distribution to \(\mathcal {G}\) . It can be derived from a result in (Benjamini et al., Advances in Mathematics 2010) that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the classical model (and hence in the new model). Using our methods, we give an alternative proof that hyperfinite hereditary properties are testable with constant query complexity and constant running time in the new model. We argue that our model is natural and our meta-theorem showing constant-time CMSO testability supports this.

On the topology of no k-equal spaces

We consider the topology of real no k-equal spaces via the theory of cellular spanning trees. Our main theorem proves that the rank of the (k−2)-dimensional homology of the no k-equal space of R is equal to the number of facets in a k-dimensional spanning tree of the k-skeleton of the n-dimensional hypercube.

Counting tournament score sequences

The score sequence of a tournament is the sequence of the out-degrees of its vertices arranged in nondecreasing order. The problem of counting score sequences of a tournament with n n vertices is more than 100 years old [Quart. J. Math. 49 (1920), pp. 1–36]. In 2013 Hanna conjectured a surprising and elegant recursion for these numbers. We settle this conjecture in the affirmative by showing that it is a corollary to our main theorem, which is a factorization of the generating function for score sequences with a distinguished index. We also derive a closed formula and a quadratic time algorithm for counting score sequences.

Clustered coloring of graphs with bounded layered treewidth and bounded degree

The clustering of a graph coloring is the maximum size of monochromatic components. This paper studies colorings with bounded clustering in graph classes with bounded layeredtreewidth, which include planar graphs, graphs of bounded Euler genus, graphs embeddable on a fixed surface with a bounded number of crossings per edge, map graphs, amongst other examples. Our main theorem says that every graph with layered treewidth at most k and with maximum degree at most Δ is 3-colorable with clustering O(k19Δ37). This is the first known polynomial bound on the clustering. This greatly improves upon a corresponding result of Esperet and Joret for graphs of bounded genus.

Soliton interactions and Yang–Baxter maps for the complex coupled short‐pulse equation

AbstractThe complex coupled short‐pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so‐called self‐symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long‐time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well‐known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.

Lower bounds for regulators of number fields in terms of their discriminants

We prove inequalities that compare the regulator of a number field with the absolute value of its discriminant. We refine the ideas in Silverman’s work [15] where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors’ previous results on bounding the product of heights of relative units in a number field extension.

Phase transitions for countable state 1D SOS model with external field

We consider a 1D solid-on-solid (SOS) model with external field in which the single-spin space is the set of all integers. Then, we construct a Gibbs specification for the model and get a functional equation such that every positive solution defines an infinite volume Gibbs measure. We show that there exist infinitely many Gibbs measures for the 1D SOS model with external field for some values of parameters θ and θ 1 (the last one is responsible to the external field). Moreover, by the main theorem, we conclude that a phase transition occurs for the 1D SOS model with external field and there is no Gibbs measure for the 1D SOS model without an external field.

Sharp concentration results for heavy-tailed distributions

Abstract We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy $P(X>t) \leq{\text{ e}}^{- I(t)}$, where $I: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing function and $I(t)/t \rightarrow \alpha \in [0, \infty )$ as $t \rightarrow \infty $. Our main theorem can not only recover some of the existing results, such as the concentration of the sum of sub-Weibull random variables, but it can also produce new results for the sum of random variables with heavier tails. We show that the concentration inequalities we obtain are sharp enough to offer large deviation results for the sums of independent random variables as well. Our analyses which are based on standard truncation arguments simplify, unify and generalize the existing results on the concentration and large deviation of heavy-tailed random variables.

Numerical corroboration of Koiter’s model for all the main types of linearly elastic shells in the static case

In this paper, we corroborate the main theorems establishing the justification of Koiter’s model, established by Ciarlet and his associates, for all the types of linearly elastic shells via a set of numerical experiments.

A New Approach to Involution in Fuzzy C★-Algebra via Functional Inequality and Python Implementation

This article explores the stability of involution in fuzzy C★-algebras through the use of a functional inequality. We present an approach to obtaining an approximate involution in fuzzy C★-algebras by utilizing a fixed-point method. Moreover, for greater clarity, we implemented Python code for the main theorem.

System of fractional boundary value problems at resonance

In this paper, we investigate the existence of solutions for a system of fractional differential equations at resonance set on the interval [0, 1]. We associate a Dirichlet condition at $$t=0$$ and an integral boundary condition at $$t=1$$ . Our main existence theorem relies on the coincidence degree theory. A detailed study of the linear operators involved in the fixed point formulation is carried out. Finally, an example is included to illustrate the applicability of theoretical result.

The Helmholtz problem in slowly varying waveguides at locally resonant frequencies

This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schrödinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schrödinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in Hloc1. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.

Stochastic comparisons of largest claim amounts from heterogeneous portfolios

This paper investigates stochastic comparisons of largest claim amounts of two sets of independent or interdependent portfolios in the sense of some stochastic orders. Let random variable () with distribution function , represents the claim amount for ith risk of a portfolio. Here two largest claim amounts are compared considering that the claim variables follow a general semiparametric family of distributions having the property that the survival function is increasing in or is increasing and convex/concave in . The results obtained in this paper apply to a large class of well‐known distributions including the family of exponentiated/generalized distributions (e.g., exponentiated exponential, Weibull, gamma and Pareto family), Rayleigh distribution and Marshall–Olkin family of distributions. As a direct consequence of some main theorems, we also obtained the results for scale family of distributions. Several numerical examples are provided to illustrate the results.

Topological K-theory for discrete groups and index theory

For any countable discrete group Γ (without any further assumptions on it) we first construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map, Ktop⁎(Γ), to the periodic cyclic homology of the group algebra, HP⁎(CΓ). This morphism, called the Chern-Baum-Connes assembly map, allows in particular to give a proper and explicit formulation for a Chern-Connes pairingKtop⁎(Γ)×HP⁎(CΓ)⟶C.Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for Γ-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to Γ which is an isomorphism once tensoring with C, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to Γ and the homology group H⁎(Γ,FΓ) (where FΓ is the complex vector space freely generated by the set of elliptic elements in Γ). This last group H⁎(Γ,FΓ) identifies, by the work of Burghelea, as a direct factor of the cyclic periodic homology of the group algebra.Moreover, in this paper Ktop⁎(Γ) stands for the topological K-theory for discrete groups as originally proposed by Baum and Connes in their first paper, where wrong way functoriality in equivariant K-theory was used for its construction. As part of our results we prove that this model is indeed isomorphic to the analytic model for the left-hand side of the assembly map.Our main final theorem gives an explicit geometric index theoretical formula for the above mentioned pairing (without any further assumptions on Γ) in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles using Burghelea's computation. This gives a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions of the discrete group on manifolds, and cyclic periodic cohomology of the group algebra.

A Tseng-type algorithm for approximating zeros of monotone inclusion and J-fixed-point problems with applications

In this paper, a Halpern–Tseng-type algorithm for approximating zeros of the sum of two monotone operators whose zeros are J-fixed points of relatively J-nonexpansive mappings is introduced and studied. A strong convergence theorem is established in Banach spaces that are uniformly smooth and 2-uniformly convex. Furthermore, applications of the theorem to convex minimization and image-restoration problems are presented. In addition, the proposed algorithm is used in solving some classical image-recovery problems and a numerical example in a Banach space is presented to support the main theorem. Finally, the performance of the proposed algorithm is compared with that of some existing algorithms in the literature.

Parameterized complexity of categorical clustering with size constraints

In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a1,…,an} over Σm, where Σ is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median objective of the clustering in the Hamming norm is at most B. Fomin, Golovach, and Panolan [ICALP 2018] proved that the problem is fixed-parameter tractable for the binary case Σ={0,1}. We extend this algorithmic result to a popular capacitated clustering model, where in addition the sizes of the clusters are lower and upper bounded by integer parameters p and q, respectively. Our main theorem is that the problem is solvable in time 2O(Blog⁡B)|Σ|B⋅(mn)O(1).

Siegel Zeros, Twin Primes, Goldbach’s Conjecture, and Primes in Short Intervals

Abstract We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular, we prove for $$ \begin{align*} & \sum_{n \leq X} \Lambda(n) \Lambda(\pm n+h) \end{align*}$$an asymptotic formula that holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$, we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost all very short intervals.

Extending the Domain with Application of Four-Step Nonlinear Scheme with Average Lipschitz Conditions

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

Approximating spaces of Nagata dimension zero by weighted trees

We prove that if a metric space X has Nagata dimension zero with constant c, then there exists a dense subset of X that is 8c-bi-Lipschitz equivalent to a weighted tree. The factor 8 is the best possible if c=1; that is, if X is an ultrametric space. This yields a new proof of a result of Chan, Xia, Konjevod, and Richa. Moreover, as an application, we also obtain quantitative versions of certain metric embedding and Lipschitz extension results of Lang and Schlichenmaier. Finally, we prove a variant of our main theorem for 0-hyperbolic proper metric spaces. This generalizes a result of Gupta.