Non-vanishing Coefficients Research Articles

Clique and cycle frequencies in a sparse random graph model with overlapping communities

. A statistical network model with overlapping communities can be generated as a superposition of mutually independent random graphs of varying size. The model is parameterized by the number of nodes, the number of communities, and the joint distribution of the community size and the edge probability. This model admits sparse parameter regimes with power-law limiting degree distributions and non-vanishing clustering coefficients. This article presents large-scale approximations of clique and cycle frequencies for graph samples generated by the model, which are valid for regimes with unbounded numbers of overlapping communities. Our results reveal the growth rates of these subgraph frequencies and show that their theoretical densities can be reliably estimated from the data.

Quaternionic Speh representations

For a central division algebra D , we study a family of representations of \operatorname{GL}_{k,D} (both locally and globally), which can be viewed as analogs of the Speh representations. At the non- Archimedean places, we show that these representations support unique models of degenerate type. Globally, we show that these representations support certain non-vanishing Fourier coefficients. We also obtain some partial results regarding unique models at the Archimedean places.

Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ

Abstract Let k k be a nonnegative integer. Let K K be a number field and O K {{\mathcal{O}}}_{K} be the ring of integers of K K . Let ℓ ≥ 5 \ell \ge 5 be a prime and v v be a prime ideal of O K {{\mathcal{O}}}_{K} over ℓ \ell . Let f f be a modular form of weight k + 1 2 k+\frac{1}{2} on Γ 0 {\Gamma }_{0} (4) such that its Fourier coefficients are in O K {{\mathcal{O}}}_{K} . In this article, we study sufficient conditions that if f f has the form f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v ) f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers s i {s}_{i} , then f f is congruent to a linear combination of iterated derivatives of a single theta function modulo v v .

The disconnectedness of certain sets defined after uni-variate polynomials

We consider the set of monic real uni-variate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $d\geq 6$ and for signs of the coefficients $(+,-,+,+,\ldots ,+,+,-,+)$, we prove that the set of such polynomials having two positive, $d-4$ negative and two complex conjugate roots, is not connected. For $pos+neg\leq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.

On Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ with applications to Siegel modular forms

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms \(\bmod \, \ell \), partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto x which do not vanish \(\bmod \, \ell \). We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients \(\bmod \, \ell \) of a Siegel modular form with integral algebraic Fourier coefficients provided \(\ell \) is large enough. We also make some efforts to make this “largeness” of \(\ell \) effective.

Representation of integers by sparse binary forms

We will give new upper bounds for the number of solutions to the inequalities of the shape | F ( x , y ) | ≤ h |F(x,y)| \leq h , where F ( x , y ) F(x,y) is a sparse binary form, with integer coefficients, and h h is a sufficiently small integer in terms of the discriminant of the binary form F F . Our bounds depend on the number of non-vanishing coefficients of F ( x , y ) F(x,y) . When F F is “really sparse”, we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in [Trans. Amer. Math. Soc. 303 (1987), pp. 241–255], [Acta Math. 160 (1988), pp. 207–247], in special but important cases.

Compression of Spin-Adapted Multiconfigurational Wave Functions in Exchange-Coupled Polynuclear Spin Systems.

We present a protocol based on unitary transformations of molecular orbitals to reduce the number of nonvanishing coefficients of spin-adapted configuration interaction expansions. Methods that exploit the sparsity of the Hamiltonian matrix and compactness of its eigensolutions, such as the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm in its spin-adapted implementation, are well suited to this protocol. The wave function compression resulting from this approach is particularly attractive for antiferromagnetically coupled polynuclear spin systems, such as transition-metal cubanes in biocatalysis, and Mott and charge-transfer insulators in solid-state physics. Active space configuration interaction calculations on N2 and CN– at various bond lengths, the stretched square N4 compounds, the chromium dimer, and a [Fe2S2]2– model system are presented as a proof-of-concept. For the Cr2 case, large and intermediate bond distances are discussed, showing that the approach is effective in cases where static and dynamic correlations are equally important. The [Fe2S2]2– case shows the general applicability of the method.

A nonrealization theorem in the context of Descartes' rule of signs

For a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$), Descartes' rule of signs says that $P$ has $pos\leq c$ positive and $neg\leq p$ negative roots, where $pos\equiv c($ mod $2)$ and $neg\equiv p($ mod $2)$. For $1\leq d\leq 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(pos, neg)$ satisfying these conditions there exists a polynomial $P$ with exactly $pos$ positive and $neg$ negative roots (all of them simple); that is, all these cases are realizable. This is not true for $d\geq 4$, yet for $4\leq d\leq 8$ (for these degrees the exhaustive answer to the question of realizability is known) in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\geq 4$. For $d=9$, we show a counterexample to this conjecture: for the sign pattern $(+,-,-,-,-,+,+,+,+,-)$ and the pair $(1,6)$ there exists no polynomial with $1$ positive, $6$ negative simple roots and a complex conjugate pair and, up to equivalence, this is the only case for $d=9$.

Nonreciprocal thermal and thermoelectric transport of electrons in noncentrosymmetric crystals

Nonreciprocal transport phenomena indicate that the forward and backward flows differ, and are attributed to broken inversion symmetry. In this paper, we study the nonreciprocity of a thermal and thermoelectric transport of electronic systems resulting from inversion-symmetry-broken crystal structures. The nonlinear electric, thermoelectric, and thermal conductivities are derived up to the second order in an electric field and a temperature gradient by using the Boltzmann equation with the relaxation time approximation. All the second-order conductivities appearing in this paper are described by two functions and their derivatives, and these are related to each other in the same way that linear conductivities are e.g. via the Wiedemann-Franz law. We found that non-vanishing thermal-transport coefficients in the zero-temperature limit appear in nonlinear conductivities, which dominate the thermal transport at a sufficiently low temperature. The nonlinear conductivities and possible observable quantities are estimated in a $1H$ monolayer of the transition metal dichalcogenides MoS$_2$ and a polar semiconductor BiTeX(X=I,Br).

Non-vanishing Fourier coefficients of Δk

Let Δk be the unique normalized cusp form of level one and weight k with k=16,18,20,22,26. In this paper, we describe a method to achieve the explicit bounds Bk of n such that the Fourier coefficients an(Δk) ≠ 0 for all n < Bk.

English

By Descartes' rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients, with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $pos\leq c$ positive and $neg\leq p$ negative roots, where $pos\equiv c($\, mod $2)$ and $neg\equiv p($\, mod $2)$. For $1\leq d\leq 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(pos, neg)$ satisfying these conditions there exists a polynomial $P$ with exactly $pos$ positive and exactly $neg$ negative roots (all of them simple). For $d\geq 4$ this is not so. It was observed that for $4\leq d\leq 10$, in all nonrealizable cases either $pos=0$ or $neg=0$. It was conjectured that this is the case for any $d\geq 4$. We show a counterexample to this conjecture for $d=11$. Namely, we prove that for the sign pattern $(+,-,-,-,-,-,+,+,+,+,+,-)$ and the pair $(1,8)$ there exists no polynomial with $1$ positive, $8$ negative simple roots and a complex conjugate pair.

On local equivalence of the majorant of partial sums and Paley function of Franklin series

In this paper we prove that the majorant of partial sums and the Paley function of Franklin series have equivalent norms in the space L p (I), p > 0, provided that the “peak” intervals of Franklin functions with non-vanishing coefficients lie in I. Examples of series emphasizing that this condition is essential are also given.

A variant of the Bombieri–Vinogradov theorem in short intervals and some questions of Serre

AbstractWe generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modularL-functionL(s, f), the fundamental discriminantsdfor which thed-quadratic twist ofL(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

Coefficients of half-integral weight modular forms and indivisibility of class numbers of imaginary quadratic fields

In this paper, we investigate the nonvanishing coefficients of half-integral weight modular forms modulo a prime. We also consider the indivisibility of class numbers of imaginary quadratic fields.

Stochastic Stability via Lyapunov Functions without Differentiability at Supposed Equilibria

Abstract: The objective of this paper is to tackle point stability of stochastic systems without requiring noise diffusion coefficients to vanish at the point of interest. Behavioral differences between vanishing coefficients and non-vanishing coefficients arising in adding stochastic noises are illustrated by examples. To identify the differences appropriately, this paper examines the concept of equilibria and proposes several stability properties by introducing the notions of instantaneous points and almost sure equilibria. In addition to clarifying the relationship between those stability properties, Lyapunov-type characterizations are presented as sufficient conditions for those stability properties by making use of functions which are not necessarily twice differentiable at the point of interest. Discussion is also given to relate the stability property of an instantaneous point with noise-to-state stability.

Taylor series of conformal mappings onto symmetric quadrilaterals

We discuss the positivity (or lack thereof) of the non-vanishing coefficients of the Taylor series expansion about for the Riemann map of a family of rectangles and rhombi with defined symmetries.

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 2: Kernel separability, space extension, and, series solution via telescopic matrices

This work focuses on the Kronecker power series solution of the explicit conical ODEs. This means that the Kronecker power series of the descriptive function vector of the ODEs has only zeroth, first and second Kronecker powers of the unknowns hence the only nonvanishing matrix coefficients are \({\mathbf {F}}_0, {\mathbf {F}}_1\) and \({\mathbf {F}}_2\). We focus on the cases where \({\mathbf {F}}_0\) also vanishes. These enable us to get and solve a two block term recursive ODE and the accompanying initial conditions. The resulting Kronecker power series’ kernel can be expressed as a binary product whose first factor which in square matrix type and a second factor which is in purely rectangular matrix algebraic structure. The constancy adding space extension separates the temporal behavior of the kernel in a scalar first factor while the second factor is again in rectangular matrix structure. We also show that the definition and use of rectangular eigenvalue problem takes us to constant solution of the original ODEs.

On adding a variable to a Frobenius manifold and generalizations

Let \(\pi :V\rightarrow M\) be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure \((\circ _{M},e_{M},g_{M})\) and typical fiber has the structure of a Frobenius algebra \((\circ _{V},e_{V},g_{V})\). Using a connection \(D\) on the bundle \(\pi : V{\,\rightarrow \,}M\) and a morphism \(\alpha :V\rightarrow TM\), we construct an almost Frobenius structure \((\circ , e_{V},g)\) on the manifold \(V\) and we study when it is Frobenius. In particular, we describe all (real) positive definite Frobenius structures on \(V\) obtained in this way, when \(M\) is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting, we add a real structure \(k_{M}\) on \(M\) and a real structure \(k_{V}\) on the bundle \(\pi : V \rightarrow M\). Using \(k_{M}\), \(k_{V}\) and \(D\) we define a real structure \(k\) on the manifold \(V\). We study when \(k\), together with an almost Frobenius structure \((\circ , e_{V}, g) \), satisfies the tt*- equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and \(tt^{*}\)-geometry.

Fourier coefficients of automorphic forms, character variety orbits, and small representations

We consider the Fourier expansions of automorphic forms on general Lie groups, with a particular emphasis on exceptional groups. After describing some principles underlying known results on GL ( n ) , Sp ( 4 ) , and G 2 , we perform an analysis of the expansions on split real forms of E 6 and E 7 where simplifications take place for automorphic realizations of real representations which have small Gelfand–Kirillov dimension. Though the character varieties are more complicated for exceptional groups, we explain how the nonvanishing Fourier coefficients for small representations behave analogously to Fourier coefficients on GL ( n ) . We use this mechanism, for example, to show that the minimal representation of either E 6 or E 7 never occurs in the cuspidal automorphic spectrum. We also give a complete description of the internal Chevalley modules of all complex Chevalley groups – that is, the orbit decomposition of the Levi factor of a maximal parabolic on its unipotent radical. This generalizes classical results on trivectors and in particular includes a full description of the complex character variety orbits for all maximal parabolics. The results of this paper have been applied in the string theory literature to the study of BPS instanton contributions to graviton scattering (Green et al., 2011, [12] ). For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ELkyOT8c28I . Author Video Watch what authors say about their articles

ON THE DISTRIBUTION OF -FREE NUMBERS AND NON-VANISHING FOURIER COEFFICIENTS OF CUSP FORMS

AbstractWe study properties of -free numbers, that is numbers that are not divisible by any member of a set . First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of -free numbers and their distribution on almost all short intervals. Results on -free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.