Regular Partition Research Articles

$$\ell $$-Adic properties and congruences of $$\ell $$-regular partition functions

$$\ell $$-Adic properties and congruences of $$\ell $$-regular partition functions

A Parity Result for Some p -Regular Partitions

Let p > 2 be prime and r ∈ { 1 , 2 , … , p − 1 } . Denote by c p ( n ) the number of p -regular partitions of n in which parts can occur not more than three times. We prove the following: If 8 r + 1 is a quadratic non-residue modulo p , c p ( p n + r ) ≡ 0 ( mod 2 ) for all nonnegative integers n .

Congruence Properties Modulo Powers of $2$ for $4$-Regular Partitions

Congruence Properties Modulo Powers of $2$ for $4$-Regular Partitions

Efficient polynomial-time approximation scheme for the genus of dense graphs

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that | E ( G )| ≥ α | V ( G )| 2 for some fixed α > 0. While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given ε > 0, returns an integer g such that G has an embedding in a surface of genus g , and this is ε-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G) \) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G) \) . The running time of the algorithm is O ( f (ε) n 2 ), where f (·) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time O ( f 1 (ε) n 2 ). Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [25]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [17] and to Fox, Lovász, and Zhao [14, 15]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ε-approximation ν ( G ) for the maximum number of edge-disjoint triangles in G . The value ν ( G ) can be computed by solving a linear program whose size is bounded by certain value f 2 (ε) depending only on ε. After solving the linear program, the genus can be approximated (see Corollary). The proof of this result is long and will be of independent interest in topological graph theory.

Unimodality of regular partition polynomials

Unimodality of regular partition polynomials

Equivalent regular partitions of three‐uniform hypergraphs

AbstractThe regularity method was pioneered by Szemerédi for graphs and is an important tool in extremal combinatorics. Over the last two decades, several extensions to hypergraphs were developed which were based on seemingly different notions of quasirandom hypergraphs. We consider the regularity lemmata for three‐uniform hypergraphs of Frankl and Rödl and of Gowers, and present a new proof that the concepts behind these approaches are equivalent.

Partition Entropy as a Measure of Regularity of Music Scales

The entropy of the partition generated by an n-tone music scale is proposed to quantify its regularity. The normalized entropy relative to a regular partition and its complementary, here referred to as the bias, allow us to analyze various conditions of similarity between an arbitrary scale and a regular scale. Interesting particular cases are scales with limited bias because their tones are distributed along specific interval fractions of a regular partition. The most typical case in music concerns partitions associated with well-formed scales generated by a single tone h. These scales are maximal even sets that combine two elementary intervals. Then, the normalized entropy depends on each number of intervals as well as their relative size. When well-formed scales are refined, several nested families stand out with increasing regularity. It is proven that a scale of minimal bias, i.e., with less bias than those with fewer tones, is always a best rational approximation of log2h.

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Testing versus estimation of graph properties, revisited

AbstractA graph on vertices is ‐far from property if one should add/delete at least edges to turn into a graph satisfying . A distance estimator for is an algorithm that given and distinguishes between the case that is ‐close to and the case that is ‐far from . If has a distance estimator whose query complexity depends only on , then is said to be estimable. Every estimable property is clearly also testable, since testing corresponds to estimating with . A central result in the area of property testing is the Fischer–Newman theorem, stating that an inverse statement also holds, that is, that every testable graph property is in fact estimable. The proof of Fischer and Newman was highly ineffective, since it incurred a tower‐type loss when transforming a testing algorithm for into a distance estimator. This raised the natural problem, studied recently by Fiat–Ron and by Hoppen–Kohayakawa–Lang–Lefmann–Stagni, whether one can find a transformation with a polynomial loss. We obtain the following results. We show that if 𝒫 is hereditary, then one can turn a tester for 𝒫 into a distance estimator with an exponential loss. This is an exponential improvement over the result of Hoppen et al., who obtained a transformation with a double exponential loss. We show that for every 𝒫, one can turn a testing algorithm for 𝒫 into a distance estimator with a double exponential loss. This improves over the transformation of Fischer–Newman that incurred a tower‐type loss. Our main conceptual contribution in this work is that we manage to turn the approach of Fischer–Newman, which was inherently ineffective, into an efficient one. On the technical level, our main contribution is in establishing certain properties of Frieze–Kannan Weak Regular partitions that are of independent interest.

Infinite families of congruences modulo $2$ for $(\ell, k)$-regular partitions

Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and modulo $12$ for $(4, 9)$-regular partition has been studied. In this paper, we use theta function identities and Newman results to prove some infinite families of congruences modulo $2$ for $(2, 7)$, $(5, 8)$, $(4, 11)$-regular partition and modulo $4$ for $(4, 5)$-regular partition.

Police and robber game on infinite chessboard

It is considered two variants of the game "Policeman and a robber", on an infinite chessboard that is a graph giving a regular partition of the plane into squares. Heuristic and precise definitions of the concepts "the initial state is winning for the pursuer", and "the initial state is winning for the evader" are formulated. Then, criteria for determining if a given initial state is winning for the pursuer or for the evader is given.

Some congruences for $ \ell $-regular partitions with certain restrictions

<abstract><p>Let $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $ denote the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct odd parts and the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct even parts, respectively. Our first goal in this note was to prove two congruence relations for $ {\rm{pod}}_\ell(n) $. Furthermore, we found a formula for the action of the Hecke operator on a class of eta-quotients. As two applications of this result, we obtained two infinite families of congruence relations for $ {\rm pod}_5(n) $. We also proved a congruence relation for $ {\rm{ped}}_\ell(n) $. In particular, we established a congruence relation modulo 2 connecting $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $.</p></abstract>

Regular Composite Resource Partitioning and Reconfiguration in Open Systems

We consider the problem of resource provisioning for real-time cyber-physical applications in an open system environment where there does not exist a global resource scheduler that has complete knowledge of the real-time performance requirements of each individual application that shares the resources with the other applications. Regularity-based Resource Partition (RRP) model is an effective strategy to hierarchically partition and assign various resource slices among such applications. However, previous work on RRP model only discusses uniform resource environment, where resources are implicitly assumed to be synchronized and clocked at the same frequency. The challenge is that a task utilizing multiple resources may experience unexpected delays in non-uniform environments, where resources are clocked at different frequencies. This paper extends the RRP model to non-uniform multi-resource open system environments to tackle this problem. It first introduces a novel composite resource partition abstraction and then proposes algorithms to construct and reconfigure the composite resource partitions. Specifically, the Acyclic Regular Composite Resource Partition Scheduling (ARCRP-S) algorithm constructs regular composite resource partitions and the Acyclic Regular Composite Resource Partition Dynamic Reconfiguration (ARCRP-DR) algorithm reconfigures the composite resource partitions in the run time upon requests of partition configuration changes. Our experimental results show that compared with state-of-the-art methods, ARCRP-S can prevent unexpected resource supply shortfall and improve the schedulability up to 50%. On the other hand, ARCRP-DR can guarantee the resource supply during the reconfiguration with moderate computational overhead.

Multilocular Cystic Nephroma in Adult: About A Case Report and Review of the Literature

Background: Multicystic nephroma is a rare entity of cystic renal tumor, it is a benign non-hereditary pathology with an excellent prognosis, whose preoperative diagnosis is often difficult. Only the anatomopathological study can confirm this. Case Presentation: 54-year-old patient, having consulted for right low back pain evolving for 6 months without any other associated sign. The clinical examination objectified a slight tenderness of the right lumbar fossa. Abdominal ultrasound revealed a 6 cm multi-partitioned cystic renal mass. The uroscanner identified a right mid-renal cystic mass with multilocules containing fine and regular partitions, with exo-renal development measuring 66*74 mm classified Bosniak IIF. A resection of the entire cyst by lombotomy was performed, the histological examination of which came back in favor of a multicystic nephroma. Conclusion: The discovery of a cystic renal tumor often represents a source of uncertainty as to its potential for malignancy. The definitive diagnosis can only be made after surgical management with histological study.

Power variations in fractional Sobolev spaces for a class of parabolic stochastic PDEs

We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a fractional power $\gamma$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < \gamma - d/2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and a nontrivial $p$th order variation for $p= 2\gamma/(\gamma-d/2-r)>2$ when $r>-d/2$. More generally, suitably normalized power variations of any order satisfy a genuine law of large numbers in the first case and a degenerate limit theorem in the second case. When $r<-d/2$, the quadratic variation is given explicitly via an expression that involves the spectral zeta function, which reduces to the Riemann zeta function when $d=1$ and $D$ is an interval.

Infinite families of congruences for $$\ell $$-regular partition functions

Infinite families of congruences for $$\ell $$-regular partition functions

De Rham compatible Deep Neural Network FEM

On general regular simplicial partitions T of bounded polytopal domains Ω⊂Rd, d∈{2,3}, we construct exact neural network (NN) emulations of all lowest order finite element spaces in the discrete de Rham complex. These include the spaces of piecewise constant functions, continuous piecewise linear (CPwL) functions, the classical “Raviart–Thomas element”, and the “Nédélec edge element”. For all but the CPwL case, our network architectures employ both ReLU (rectified linear unit) and BiSU (binary step unit) activations to capture discontinuities. In the important case of CPwL functions, we prove that it suffices to work with pure ReLU nets. Our construction and DNN architecture generalizes previous results in that no geometric restrictions on the regular simplicial partitions T of Ω are required for DNN emulation. In addition, for CPwL functions our DNN construction is valid in any dimension d≥2. Our “FE-Nets” are required in the variationally correct, structure-preserving approximation of boundary value problems of electromagnetism in nonconvex polyhedra Ω⊂R3. They are thus an essential ingredient in the application of e.g., the methodology of “physics-informed NNs” or “deep Ritz methods” to electromagnetic field simulation via deep learning techniques. We indicate generalizations of our constructions to higher-order compatible spaces and other, non-compatible classes of discretizations, in particular the “Crouzeix–Raviart” elements and Hybridized, Higher Order (HHO) methods.

A partition-based convergence framework for population-based optimization algorithms

Population-based optimization algorithms, such as genetic algorithm and particle swarm optimization, have become a class of important algorithms for solving global optimization problems. However, there is an issue that the global convergence is often absent for most of them. This paper proposes a partition-based convergence framework for population-based optimization algorithms to solve this troubling problem. In this framework, regular partitions and evolutions of populations are implemented alternatively. Specifically, the initial population is generated from a regular partition on the search space; after several generations of evolution of the population, the evolution result is returned to join in the regular partition again, and a new population is generated. Repeat such progress until some stop condition is satisfied. Global convergence is guaranteed for the framework. Then this convergence framework is applied to particle swarm optimization, differential evolution, and genetic algorithm. The modified algorithms are globally convergent and perform better than the original version.

$ W^{1, \infty} $-seminorm superconvergence of the block finite element method for the five-dimensional Poisson equation

&lt;abstract&gt;&lt;p&gt;This study focused on the superconvergence of the finite element method for the five-dimensional Poisson equation in the $ W^{1, \infty} $-seminorm. Specifically, we investigated the block finite element method, which is a tensor-product finite element approach applied to regular rectangular partitions of the domain. First, we introduced the finite element scheme for the equation and discussed various functions related to it, along with their properties. Next, we proposed a weight function and established its important properties, which play a crucial role in the theoretical analysis. By utilizing the properties of the weight function and employing weighted-norm analysis techniques, we derived an optimal order estimate in the $ W^{2, 1} $-seminorm for the discrete derivative Green's function (DDGF). Furthermore, we provided an interpolation fundamental estimate of the second type, also known as the weak estimate of the second type, for the block finite element. This weak estimate is based on a five-dimensional interpolation operator of the projection type. Finally, by combining the derived $ W^{2, 1} $-seminorm estimate for the DDGF and the weak estimate for the block finite element, we obtained a superconvergence estimate for the block finite element approximation in the pointwise sense of the $ W^{1, \infty} $-seminorm. The correctness of the theoretical results was demonstrated through a numerical example.&lt;/p&gt;&lt;/abstract&gt;

-REGULAR PARTITIONS AND MODULAR FORMS WITH COMPLEX MULTIPLICATION

AbstractA partition of a positive integer n is called $\ell $ -regular if none of its parts is divisible by $\ell $ . Denote by $b_{\ell }(n)$ the number of $\ell $ -regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$ .