Arithmetic Conditions Research Articles

Equidistribution of expanding degenerate manifolds in the space of lattices

AbstractFor the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real‐analytic submanifold under a diagonal flow. This work extends the earlier result of Shah in the case of nondegenerate submanifolds. We apply the above dynamical result to show that if the affine span of a real‐analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet‐improvable.

Groups with some arithmetic conditions on real sub-class sizes

Groups with some arithmetic conditions on real sub-class sizes

Verification of Source Activations in a 3D Brain Model Using 'CLEVER' Algorithm for Mental Arithmetic Conditions.

The focus of this study is to detect the exact location of the source which causes such damage. In this article, we analyse the mental conditions of subjects under a workload of performing mental arithmetic calculations for various frequency bands and plot the topography to understand the areas of active potentials. We propose a Novel Cluster Ensemble Verifier (CLEVER) algorithm, which combines two different techniques: clustering and source localisation. The proposed algorithm is highly efficient in identifying the exact location of the source. It is seen that the topographic plots of the independent component analysis (ICA), which has the maximum percentage of relative variance, correlates to the cluster generated. We are able to give the percentage-wise contribution of every component which is responsible for brain source activation with less time complexity. Out of 72 subjects, in 67 subjects, 299 out of 433 components originate from the occipital and parietal areas of the brain with a maximum power of 43.5 µv2. As an example, the relative variance of one component is found to be contributing up to 74.03% to source activations. Clusters show similarity across the subjects in the parietal and occipital areas of the brain. The dataset used for experimentation is EEGMAT from Physionet's repository. The computation time for the algorithms is 17.6 ± 3.2 minutes. Conclusion: Findings show that during mental arithmetic calculations, both occipital and parietal areas of the brain are involved. As the data is acquired by orally mentioning the mathematical problem, subjects tend to visualise the numbers while finding the solution, which is reflected in the occipital area of the brain. CLEVER algorithm verifies the origin of the activity in the occipital and parietal areas of the brain.

Hyperplane Arrangements Satisfy (Un)Twisted Logarithmic Comparison Theorems, Applications to $\mathscr {D}_{X}$ -modules

Abstract For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$ -module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs , generalized Paley sum graphs , and their corresponding complements and , for k = 3, 4. Denote by either or . We compute the spectra of and and from them we obtain the spectra of and also. Then we show that, in the non-semiprimitive case, the spectrum of and with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs and for any , respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of such that and are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs and all strongly regular graphs which are complementary equienergetic, i.e. and are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs which are neither bipartite nor strongly regular.

Galois groups of certain even octic polynomials

Let [Formula: see text] be an irreducible polynomial where [Formula: see text] is a square. We give a method that completely describes the factorization patterns of a linear resolvent of [Formula: see text] using simple arithmetic conditions on [Formula: see text] and [Formula: see text]. As a result, we determine the exact six possible Galois groups of [Formula: see text] and completely classify all of them. As an application, we characterize the Galois groups of irreducible polynomials [Formula: see text]. We also use similar methods to obtain analogous results for the Galois groups of irreducible polynomials [Formula: see text].

A reduction formula for Waring numbers through generalized Paley graphs

We give a reduction formula for the Waring number g(k, q) over a finite field $$\mathbb {F}_q$$ . By exploiting the relation between g(k, q) with the diameter of the generalized Paley graph $$\Gamma (k,q)$$ and by using the characterization due to Pearce and Praeger (2019) of those $$\Gamma (k,q)$$ which are Cartesian decomposable, we obtain the reduction formula $$\begin{aligned} g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a) \end{aligned}$$ for p prime and a, b, c positive integers under certain arithmetic conditions. Then, we find some arithmetic conditions to apply the formula above, which allow us to obtain many infinite families of explicit values of Waring numbers. Finally, we use the reduction formula together with the characterization of 2-weight irreducible cyclic codes due to Schmidt and White (2002) to find infinite families of explicit even values of g(k, q).

On homomorphisms into Weyl modules corresponding to partitions with two parts

AbstractLet K be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda) \to \Delta (\mu)$ of Weyl modules for the general linear group $GL_n(K)$ . Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.

Arithmetic properties encoded in the Galois module structure of [formula omitted

The power classes of a field are well-known for their ability to parameterize elementary p-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple.Suppose m,n∈N and p is prime. In a recent paper, a surprising and elegant decomposition for pmth power classes has been computed when the underlying Galois group is a cyclic group of order pn. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when p=2, the structure of this exceptional summand was determined by a vector a∈{−∞,0,…,n}m and a natural number d. In this paper we give field-theoretic interpretations for a and d, showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to K/F.

Arithmetic properties of 3-cycles of quadratic maps over [formula omitted

It is shown that c=−29/16 is the unique rational number of smallest denominator, and the unique rational number of smallest numerator, for which the map fc(x)=x2+c has a rational periodic point of period 3. Several arithmetic conditions on the set of all such rational numbers c and the rational orbits of fc(x) are proved. A graph on the numerators of the rational 3-periodic points of maps fc is considered which reflects connections between solutions of norm equations from the cubic field of discriminant −23.

On the indecomposability of a remarkable new family of modules appearing in Galois theory

A powerful new perspective in the analysis of absolute Galois groups has recently emerged from the study of Galois modules related to classical parameterizing spaces of certain Galois extensions. The recurring trend in these decompositions is their stunning simplicity: almost all summands are free over some quotient ring. The non-free summands which appear are exceptional not only because they are different in form, but because they play the key role in controlling arithmetic conditions that allow the remaining summands to be easily described. In this way, these exceptional summands are the lynchpin for a bevy of new properties of absolute Galois groups that have been gleaned from these surprising decompositions.In one such recent decomposition, a remarkable new exceptional summand was discovered which exhibited interesting properties that have not been seen before. The exceptional summand is drawn from a particular finite family that has not yet been investigated. The main goal of this paper is to introduce this family of modules and verify their indecomposability. We believe this module will be of interest to people working in Galois theory, representation theory, combinatorics, and general algebra. The analysis of these modules includes some interesting new tools, including analogs of p-adic expansions.

Hairy Cantor sets

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeomorphic.Hairy Cantor sets appear in the study of the dynamics of holomorphic maps with infinitely many renormalisation structures. They are employed to link the fundamental concepts of polynomial-like renormalisation by Douady-Hubbard with the arithmetic conditions obtained by Herman-Yoccoz in the study of the dynamics of analytic circle diffeomorphisms.

Overlapping Neuronal Population Responses in the Human Parietal Cortex during Visuospatial Attention and Arithmetic Processing.

Engagement of posterior parietal cortex (PPC) in visuospatial attention and arithmetic processing has been extensively documented using neuroimaging methods. Numerous studies have suggested a close connection between visuospatial attention and arithmetic processing. However, the extant evidence in humans stems from neuroimaging methods that have relied on group analyses without much knowledge about the profile of neurophysiological engagement within localized neuronal populations at the individual brain level. Hence, it has remained unclear if the overlap of two functions in the PPC is the product of averaging, or they truly stem from a common profile of activity within the same neuronal populations in the human PPC. In the current study, we leveraged the anatomical precision and high signal-to-noise ratio of intracranial electrocorticography and probed the engagement of discrete PPC neuronal populations in seven neurosurgical patients (n = 179 total PPC sites covered; 26 sites on average per individual participant). We aimed to study the extent of parietal activations within each individual brain during visuospatial attention versus arithmetic tasks and the profile of electrophysiological responses within a given recording site during these tasks. Our findings indicated that about 40% of PPC sites did not respond to either visuospatial attention or arithmetic stimuli-or episodic memory conditions that were used as an adjunct control condition. Of those that were activated during either visuospatial attention or arithmetic conditions, a large majority showed overlapping responses during both visuospatial attention and arithmetic conditions. Most interestingly, responses during arithmetic processing were greatest in sites along the intraparietal sulcus region showing preference to contralateral, instead of ipsilateral, visual probes in the visuospatial attention task. Our results provide novel data about the relationship between numerical and spatial orientation at the neuronal population level and shed light on the complex functional organization of the PPC that could not be attained with noninvasive methods.

On existence of quasi-Strebel structures for meromorphic $k$-differentials

In this paper, motivated by the classical notion of a Strebel quadratic differential on a compact Riemann surface without boundary we introduce the notion of a quasi-Strebel structure for a meromorphic differential of an arbitrary order. It turns out that every differential of even order k\ge 4 satisfying certain natural conditions at its singular points admits such a structure. The case of differentials of odd order is quite different and our existence result involves some arithmetic conditions. We discuss the set of quasi-Stebel structures associated to a given differential and introduce the subclass of positive k -differentials. Finally, we provide a family of examples of positive rational differentials and explain their connection with the classical Heine–Stieltjes theory of linear differential equations with polynomial coefficients.

Periods of Hodge cycles and special values of the Gauss' hypergeometric function

We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by “simple” arithmetic conditions. We explore some examples and give an upper bound for the dimension of this subspace. As an application, we find explicit expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable.

On some new results for the generalised Lucas sequences

Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

Critical group structure from the parameters of a strongly regular graph

We give simple arithmetic conditions that force the Sylow p-subgroup of the critical group of a strongly regular graph to take a specific form. These conditions depend only on the parameters (v,k,λ,μ) of the strongly regular graph under consideration. We give many examples, including how the theory can be used to compute the critical group of Conway's 99-graph and to give an elementary argument that no srg(28,9,0,4) exists.

On the discrepancy of random subsequences of $\{n\alpha \}$, II

Let $\alpha$ be an irrational number, let $X_1, X_2, \ldots$ be independent, identically distributed, integer-valued random variables, and put $S_k=\sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $P (|X_1|>t)\sim ct^{-\beta}$, $0<\beta<2$, in Part I of this paper we proved that, up to logarithmic factors, the order of magnitude of the discrepancy $D_N (S_k \alpha)$ of the first $N$ terms of the sequence $\{S_k \alpha\}$ is $O(N^{-\tau})$, where $\tau= \min (1/(\beta \gamma), 1/2)$ (with $\beta=2$ in the case of finite variances) and $\gamma$ is the strong Diophantine type of $\alpha$. This shows a change of behavior of the discrepancy at $\beta\gamma=2$. In this paper we determine the exact order of magnitude of $D_N (S_k \alpha)$ for $\beta\gamma<1$, and determine the limit distribution of $N^{-1/2} D_N (S_k \alpha)$. We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence $\{S_k \alpha\}$. Finally, we extend our results to the discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic conditions on $X_1$, assuming only a mild polynomial rate on the weak convergence of $\{S_k\}$ to the uniform distribution.

Eisenstein series for rank one unitary groups and some cohomological applications

Let U/Q be a unitary group of Q-rank one so that the group of real points U(R)≅U(n,1). The group U is only quasi-split over Q if and only if n=1,2. The cohomology of a congruence subgroup of U is closely related to the theory of automorphic forms. This relation is best captured in the so-called automorphic cohomology spaces H⁎(U,C), a natural module under the action of the group U(Af). This paper gives a structural account of the U(Af)-module structure of that part of the cohomology which is generated by residues or derivatives of Eisenstein series. In particular, we determine a set of arithmetic conditions, mainly given in terms of partial automorphic L-functions, subject to which residues of Eisenstein series may give rise to non-vanishing cohomology classes. The main task is, although the usual method due to Langlands-Shahidi is not applicable, to analyze the analytic behavior of suitable Eisenstein series and to determine the location of their possible poles.

On the Iwasawa λ-invariant of the cyclotomic ℤ2-extension of ℚ(pq) and the 2-part of the class number of ℚ(pq,2 + 2)

In this paper, we study the Iwasawa [Formula: see text]-invariant of the cyclotomic [Formula: see text]-extension of [Formula: see text], where [Formula: see text] are distinct odd prime numbers satisfying certain arithmetic conditions. Moreover, we obtain an upper bound of the [Formula: see text]-part of the class number of certain quartic number fields by calculating the Sinnott index explicitly.