Search

Five-axis machining of free-form surfaces has one major advantage over three-axis machining, i.e. a greater degree of flexibility in positioning the cutting tool relative to the surface. In five-axis machining of free-form surfaces there are three major phases in creating the tool path: (i) generation of the cutter contact (CC) points; (ii) formation of the cutter location (CL) points that define the path followed by the cutting tool reference point; and (iii) the creation of the specific machine tool part program (G-code file). In this paper, the free-form surface is defined as a triangular polyhedral mesh. The CC-points are created from the surface mesh definition employing a cutting plane technique. Additionally, the CC points are positioned based on an examination of several important factors: geometric constraints derived from the machine tool axis limits, gouging (undercutting) of the free-form surface by the cutting tool and collision of the tool with either the machining stock or machine. The CL points can then be generated along the resulting CC points by consideration of specific machining strategies, such as cusp height and a smooth change in tool posture. The critical issues addressed in this work concern the avoidance of machining problems (machine limit, collision, and gouging) in the tool-path generation phase. Therefore, this technique avoids inefficient five-axis machining practices by automatically creating and verifying a feasible tool-path prior to the actual metal cutting. [ABSTRACT FROM AUTHOR]

Abstract In this work, we study various properties of two families of normal subgroups of the Hecke group Γ 0 (3) and their associated modular curves and function fields. As consequences, we obtain the stabilizer subgroups of the N -th roots of a uniformizer for X (Γ 0 (3)) , and realize such N -th roots as uniformizers for the associated modular curves to their stabilizer groups. [ABSTRACT FROM AUTHOR]

We analyze different ways of constructing binary extended formulations of polyhedral mixed-integer sets with bounded integer variables and compare their relative strength with respect to split cuts. We show that among all binary extended formulations where each bounded integer variable is represented by a distinct collection of binary variables, what we call “unimodular” extended formulations are the strongest. We also compare the strength of some binary extended formulations from the literature. Finally, we study the behavior of branch-and-bound on such extended formulations and show that branching on the new binary variables leads to significantly smaller enumeration trees in some cases. [ABSTRACT FROM AUTHOR]

The denominator formula for the Monster Lie algebra is the product expansion for the modular function J ( z ) − J ( τ ) given in terms of the Hecke system of SL 2 ( Z ) -modular functions j n ( τ ) . It is prominent in Zagier's seminal paper on traces of singular moduli, and in the Duncan–Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the j n ( z ) as a weight 2 modular form with a pole at z . Although these results rely on the fact that X 0 ( 1 ) has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the X 0 ( N ) modular curves. We use these functions to study divisors of modular forms. [ABSTRACT FROM AUTHOR]

An invariant random subgroup H ≤ G is a random closed subgroup whose law is invariant to conjugation by all elements of G. When G is locally compact and second countable, we show that for every invariant random subgroup H ≤ G there almost surely exists an invariant measure on G/H. Equivalently, the modular function of H is almost surely equal to the modular function of G, restricted to H. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups. [ABSTRACT FROM AUTHOR]

In this paper, we study two families of normal subgroups of Γ 0 ( 2 ) with abelian quotients and their associated modular curves. They are similar to Fermat groups and Fermat curves in some aspects but very different in other aspects. Almost all of them are non-congruence subgroups. These modular curves are either projective lines or hyperelliptic curves. There are few modular forms of weight 1 for these groups. We also determine their cuspidal divisor class groups and show that these groups are finite. [ABSTRACT FROM AUTHOR]

We prove the existence of fixed points of monotone-contraction mappings in modular function spaces. This is the modular version of the Ran and Reurings fixed point theorem. We also discus the extension of these results to the case of pointwise monotone-contraction mappings in modular function spaces. [ABSTRACT FROM AUTHOR]

Let C be a ρ-bounded, ρ-closed, convex subset of a modular function space [InlineEquation not available: see fulltext.]. We investigate the problem of constructing common fixed points for asymptotic pointwise nonexpansive semigroups of mappings [InlineEquation not available: see fulltext.], i.e. a family such that [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], and [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.], for every [InlineEquation not available: see fulltext.]. MSC: 47H09, 46B20, 47H10, 47E10. [ABSTRACT FROM AUTHOR]

In Andersen and Ueno (J Knot Theory Ramif 16:127-202, ) we constructed the vacua modular functor based on the sheaf of vacua theory developed in Tsuchiya et al. (Adv Stud Pure Math 19:459-566, ) and the abelian analog in Andersen and Ueno (Int J Math 18:919-993, ). We here provide an explicit isomorphism from the modular functor underlying the skein-theoretic model for the Witten-Reshetikhin-Turaev TQFT due to Blanchet, Habbeger, Masbaum and Vogel to the vacua modular functor. This thus provides a geometric construction of the TQFT first proposed by Witten and constructed first by Reshetikhin-Turaev from the quantum group $$U_q(\text{ sl }(N))$$ . [ABSTRACT FROM AUTHOR]

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, combining the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular, we consider the case when the modular function is multiplicatively integrable, i.e., when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions. We apply this procedure to the case of a family of Poisson structures on $${\mathbb{C}P_n}$$ , seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU( n + 1). We show that a bihamiltonian system on $${\mathbb{C}P_n}$$ defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu's description of quantum homogeneous spaces as groupoid C*-algebras. [ABSTRACT FROM AUTHOR]

Let ${\rm F}$ be a non-Archimedean locally compact field of residue characteristic $p$, let ${\rm D}$ be a finite-dimensional central division ${\rm F}$-algebra and let $\ell$ be a prime number different from $p$. We develop a theory of $\ell$-modular types for the group ${\rm GL}_m({\rm D})$, $m\geqslant 1$, in preparation of the study of the $\ell$-modular smooth representations of this group. [ABSTRACT FROM PUBLISHER]

We discuss a polyhedral embedding of the classical Fricke-Klein regular map of genus 5 in ordinary space E3. This polyhedron was originally discovered by Grünbaum in 1999, but was recently rediscovered by Brehm andWills. We establish isomorphism of the Grünbaum polyhedron with the Fricke-Klein map, and confirm its combinatorial regularity. The Grünbaum polyhedron is among the few currently known geometrically vertex-transitive polyhedra of genus g ≥ 2, and is conjectured to be the only vertex-transitive polyhedron in this genus range that is also combinatorially regular. We also contribute a new vertex-transitive polyhedron, of genus 11, to this list, as the 7th known example. In addition we show that there are only finitely many vertex-transitive polyhedra in the entire genus range g ≥ 2. [ABSTRACT FROM AUTHOR]

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigencusp forms for the full modular group SL(ℤ). We also prove a similar kind of quantitative result for the number of sign changes of the q-exponents $c(p)~(p \mbox{~vary ~over ~primes})$ of certain generalized modular functions for the congruence subgroup Γ( N), where N is square-free. [ABSTRACT FROM AUTHOR]

For an odd integer N, we study the action of Atkin's U(2)-operator on the modular function x(t) associated to the Fermat curve: XN + YN = 1. The function x(t) is modular for the Fermat group φ(N), generically a noncongruence subgroup. If x(t) = q-1+Σ∞i=1 a(iN-1)qiN-1, we essentially prove that limn→0 a(n) = 0 in the 2-adic topology. [ABSTRACT FROM AUTHOR]

Modular equations and generalized Lambert series are given for theta functions G1(q) and H1(q). [ABSTRACT FROM AUTHOR]

After Zagier proved that the traces of singular moduli are Fourier coefficients of a weakly holomorphic modular form, various arithmetic properties of the traces of singular values of modular functions mostly on the full modular group have been found. The purpose of this paper is to generalize the results for modular functions on congruence subgroups with arbitrary level. [ABSTRACT FROM AUTHOR]

Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL2(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = ∑n≥0bn tn, bn ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion ∑γnqn, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if $f\frac{q}{t}\frac{dt}{dq} = \sum a_i\ell(d_i z)$ for some integers ai, di, then the congruence relation bmpr -γpbmpr-1 + εppk+1bmpr-2 ≡ 0 (mod pr) holds. We give a number of examples of this phenomena. [ABSTRACT FROM AUTHOR]

We analyse and compare the complexity of several algorithms for computing modular polynomials. Under the assumption that rounding errors do not influence the correctness of the result, which appears to be satisfied in practice, we show that an algorithm relying on floating point evaluation of modular functions and on interpolation has a complexity that is up to logarithmic factors linear in the size of the computed polynomials. In particular, it obtains the classical modular polynomial $Phi _ell $ of prime level~$ell $ in time [ O left (ell ^2 log ^3 ell M (ell ) right ) subseteq O left ( ell ^3 log ^{4 + varepsilon }ell right ), ] where $M(ell )$ is the time needed to multiply two $ell $-bit numbers. par Besides treating modular polynomials for $Gamma ^0 (ell )$, which are an important ingredient in many algorithms dealing with isogenies of elliptic curves, the algorithm is easily adapted to more general situations. Composite levels are handled just as easily as prime levels, as well as polynomials between a modular function and its transform of prime level, such as the Schläfli polynomials and their generalisations. par Our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around $10000$ in less than two weeks on ten processors. [ABSTRACT FROM AUTHOR]

New properties of P-sets, which constitute a large class of convex compact sets in ℝ n that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ℝ n is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ⊂ ℝ n , there exists a set A 0 ⊂ A trapped in M and containing at most 2 n elements. An example of a convex compact set M ⊂ ℝ n for which such a finite set A 0 ⊂ A does not exist is given. [ABSTRACT FROM AUTHOR]

Abstract: Properties of complex networks, such as small-world property, power-law degree distribution, network transitivity, and network- community structure which seem to be common to many real-world networks have attracted great interest among researchers. In this study, global information of the networks is considered by defining the profile of any node based on the shortest paths between it and all the other nodes in the network; then a useful iterative procedure for community detection based on a measure of information discrepancy and the popular modular function is presented. The new iterative method does not need any prior knowledge about the community structure and can detect an appropriate number of communities, which can be hub communities or non-hub communities. The computational results of the method on real networks confirm its capability. [Copyright &y& Elsevier]

Abstract: In this paper we establish a relationship between the basic subdifferential and upper exhausters of positively homogeneous and polyhedral functions. In the case of a finite exhauster this relationship is represented in a form of an equality, and in the case of a Lipschitz function an inclusion formula is obtained. [Copyright &y& Elsevier]

Abstract: In this paper, the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions, such as and . Making use of these results, the authors obtain some sharp inequalities for the so-called Ramanujan''s generalized modular functions. [Copyright &y& Elsevier]

We give a geometric construct of a modular functor for any simple Lie-algebra and any level by twisting the constructions in [16, 19] by a certain fractional power of the abelian theory first considered in [13] and further studied in [2]. [ABSTRACT FROM AUTHOR]

Abstract: We determine the 2-modular character table of the Fischer group . [Copyright &y& Elsevier]

The article throws light on the direct extension theorem. By considering H and K as finite groups and by considering an extension 1 → H → G → K → 1, the paper mainly emphasizes on the special case of G ≅ H × K. It is considered in general that the isomorphism class of G does not determine extension.

Restrictions of irreducible p -restricted representations of algebraic groups of type A n to naturally embedded subgroups of type A 2 are studied. Let n> 2 and ? = S n i=1 m i ? i be the highest weight of the representation considered. The composition factors of such restrictions are determined in the cases where either n> 5 and S n i=1 m i< p , or n> 3 and m i + m i +1 + m i +2 + m i +3< p - 2 for odd i with 1 = i = and m n-j- 3 + m n-j- 2 + m n-j- 1 + m n-j< p - 2 for even j with 0 = j = . For arbitrary p -restricted representations some large composition factors are found. [ABSTRACT FROM AUTHOR]

We give a characterization of those meromorphic modular functions on a subgroup of finite index of the full modular group whose divisors are supported at the cusps, in terms of the growth of the exponents of their infinite product expansions. [ABSTRACT FROM AUTHOR]

The Kappa function is introduced as the function κ satisfying J(κ(τ)) = λ(τ), where J and λ are the elliptic modular functions. A Fourier expansion of κ is studied. [ABSTRACT FROM AUTHOR]

Let $M$ be the Hardy–Littlewood maximal function and let $ M_{(\alpha)}$ be the $\ell^\alpha$-maximal operator defined by $$M_{(\alpha)} \bar{f}(x)=\|{M\bar {f}\|_{\ell^\alpha}=\Bigg(\sum_{i=1}^\infty Mf_i(x)^\alpha\Bigg)^{1/\alpha},$$ where $\bar f=(f_i)_i$, $M\bar f=(Mf_i)_i$ and $\alpha >1$. The purpose of this work is to study modular inequalities of the form $$\int_{{\mathbb R}^n} P \big(\big|\overline M_{(\alpha)}f(x)\big| \big)\,dx \le \sum_j\int_{{\mathbb R}^n } Q(|f_j(x)|)\, dx,$$ where $P$ and $Q$ are modular functions. Our results apply to operators of the form $$\overline T_{(\alpha)} \bar f(x)= \Bigg(\sum_{i=1}^\infty |T_i f_i(x)|^\alpha\Bigg)^{1/\alpha},$$ where $T_i$ satisfies similar properties to $M$. [ABSTRACT FROM AUTHOR]

We give a new, simpler proof that the canonical actions of finite groups on Fricke-type Monstrous Lie algebras yield genus zero functions in generalized Monstrous Moonshine, using a Borcherds–Kac–Moody Lie algebra decomposition due to Jurisich. We describe a compatibility condition, arising from the no-ghost theorem in bosonic string theory, that yields the genus zero property. We give evidence for and against the conjecture that such a compatibility for symmetries of the Monster Lie algebra gives a characterization of the Monster group. [ABSTRACT FROM AUTHOR]

This paper establishes the result that numbers of the shape 5[sup 2]p(p ≡ 2 mod9)and 5[sup 2]p[sup 2](p ≡ 5 mod9), where pa prime, are expressible as the sum of two rational cubes by finding modular parametrizations of the corresponding elliptic curves, picking special complex points on the curve and manipulating them to produce rational points thereon. [ABSTRACT FROM AUTHOR]