Nontrivial Linear Combination Research Articles

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

AbstractLet be a random matrix distributed according to uniform probability measure on the finite general linear group . We show that equidistributes on as as long as and that this range is sharp. We also show that nontrivial linear combinations of equidistribute as long as and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for , where depends on , due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of , we end up showing that certain explicit character sums modulo exhibit cancellation when averaged over monic polynomials of degree in as long as . This goes far beyond the classical range due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

Expansivity and Periodicity in Algebraic Subshifts

A d-dimensional configuration c:mathbb {Z}^dlongrightarrow A is a coloring of the d-dimensional infinite grid by elements of a finite alphabet Asubseteq mathbb {Z}. The configuration c has an annihilator if a non-trivial linear combination of finitely many translations of c is the zero configuration. Writing c as a d-variate formal power series, the annihilator is conveniently expressed as a d-variate Laurent polynomial f whose formal product with c is the zero power series. More generally, if the formal product is a strongly periodic configuration, we call the polynomial f a periodizer of c. A common annihilator (periodizer) of a set of configurations is called an annihilator (periodizer, respectively) of the set. In particular, we consider annihilators and periodizers of d-dimensional subshifts, that is, sets of configurations defined by disallowing some local patterns. We show that a (d-1)-dimensional linear subspace Ssubseteq mathbb {R}^d is expansive for a subshift if the subshift has a periodizer whose support contains exactly one element of S. As a subshift is known to be finite if all (d-1)-dimensional subspaces are expansive, we obtain a simple necessary condition on the periodizers that guarantees finiteness of a subshift or, equivalently, strong periodicity of a configuration. We provide examples in terms of tilings of mathbb {Z}^d by translations of a single tile.

One-Bipolar Topologically Slice Knots and Primary Decomposition

Abstract Let $\{{\mathcal{T}}_n\}$ be the bipolar filtration of the smooth concordance group of topologically slice knots, which was introduced by Cochran et al. It is known that for each $n\ne 1$ the group ${\mathcal{T}}_n/{\mathcal{T}}_{n+1}$ has infinite rank and ${\mathcal{T}}_1/{\mathcal{T}}_2$ has positive rank. In this paper, we show that ${\mathcal{T}}_1/{\mathcal{T}}_2$ also has infinite rank. Moreover, we prove that there exist infinitely many Alexander polynomials $p(t)$ such that there exist infinitely many knots in ${\mathcal{T}}_1$ with Alexander polynomial $p(t)$ whose nontrivial linear combinations are not concordant to any knot with Alexander polynomial coprime to $p(t)$, even modulo ${\mathcal{T}}_2$. This extends the recent result of Cha on the primary decomposition of ${\mathcal{T}}_n/{\mathcal{T}}_{n+1}$ for all $n\ge 2$ to the case $n=1$. To prove our theorem, we show that the surgery manifolds of satellite links of $\nu ^+$-equivalent knots with the same pattern link have the same Ozsváth–Szabó $d$-invariants, which is of independent interest. As another application, for each $g\ge 1$, we give a topologically slice knot of concordance genus $g$ that is $\nu ^+$-equivalent to the unknot.

The smallest upper bound on the number of zeros of Abelian integrals

In this paper, we develop a new method to estimate the smallest upper bound on the number of isolated zeros of Abelian integrals, and give an algebraic criterion for the case of Abelian integrals along energy level ovals of potential systems. As applications of our main result, we first obtain a criterion guaranteeing that two Abelian integrals have the Chebyshev property with accuracy one and give an example of hyperelliptic Abelian integrals. Then we get a criterion that any nontrivial linear combination of three Abelian integrals has at most two isolated zeros. Using this criterion we prove that the smallest upper bound is two for the number of isolated zeros of a non-algebraic Abelian integral along the oval (y2+12)1x2+ln⁡|x|−12=h with 0<h<+∞, which corresponds to the perturbation of a quadratic reversible system with two centers. Moreover, we derive all possible configurations of limit cycles from the Poincaré bifurcation of this perturbation. To our knowledge, these problems can not be solved by other known approaches.

A theorem of Brown–Halmos type for dual truncated Toeplitz operators

In this paper, we investigate commuting dual truncated Toeplitz operators on the orthogonal complement of the model space $$K^{2}_{u}.$$ Let $$f,g \in L^{\infty },$$ if two dual truncated Toeplitz operators $$D_{f}$$ and $$D_{g}$$ commute, we obtain similar conditions of Brown–Halmos Theorem for Hardy-Toeplitz operators, that is, both f and g are analytic, or both f and g are co-analytic, or a nontrivial linear combination of f and g is constant. However, the first two conditions are not sufficient, one can easily construct two non-commuting dual truncated Toeplitz operators with analytic or co-analytic symbols. We prove that two bounded dual truncated Toeplitz operators $$D_{f}$$ and $$D_{g}$$ commute if and only if f, g, $${\bar{f}}(u-\lambda )$$ and $${\bar{g}}(u-\lambda )$$ all belong to $$H^{2}$$ for some constant $$\lambda ;$$ or $${\bar{f}},{\bar{g}}$$, $$f(u-\lambda )$$ and $$g(u-\lambda )$$ all belong to $$H^{2}$$ for some constant $$\lambda ;$$ or a nontrivial linear combination of f and g is constant.

Asymptotic Rank Theorems

Let A be a numerical k ×∞-matrix such that minors AI of order k tend to zero if numbers of all columns forming these minors tend to infinity. It is shown that there exits a nontrivial linear combination of rows in A which is a sequence tending to zero.

Semigroups of $L$-space knots and nonalgebraic iterated torus knots

Algebraic knots are known to be iterated torus knots and to admit L-space surgeries. However, Hedden proved that there are iterated torus knots that admit L-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots in this family is concordant to a linear combination of algebraic knots. The proof uses the Ozsvath-Stipsicz-Szabo Upsilon function, and also introduces a new invariant of L-space knots, the formal semigroup.

Numerical investigation of polarization filtering for direct optical imaging within scattering media.

We investigate the ability of polarization filtering to improve direct imaging of absorbing objects which are buried within scattering environments. We extend on previous empirical investigations by exploiting an efficient perturbation-based formalism, which is applicable to arbitrarily arranged sources and detectors with arbitrary polarizations. From this approach, we are able in some cases to find certain non-trivial linear combinations of polarization measurement channels that maximize the object resolution and visibility.

Properties of the single-site reduced density matrix in the Bose-Bose resonance model in the ground state and in quantum quenches

We study properties of the single-site reduced density matrix in the Bose-Bose resonance model as a function of system parameters. This model describes a single-component Bose gas with a resonant coupling to a diatomic molecular state, here defined on a lattice. A main goal is to demonstrate that the eigenstates of the single-site reduced density matrix have structures that are characteristic for the various quantum phases of this system. Since the Hamiltonian conserves only the global particle number but not the number of bosons and molecules individually, these eigenstates, referred to as optimal modes, can be nontrivial linear combinations of bare eigenstates of the molecular and boson particle number. We numerically analyze the optimal modes and their weights, the latter giving the importance of the corresponding state, in the ground state of the Bose-Bose resonance model. We find that the single-site von Neumann entropy is sensitive to the location of the phase boundaries. We explain the structure of the optimal modes and their weight spectra using perturbation theory and via a comparison to results for the single-component Bose-Hubbard model. We further study the dynamical evolution of the optimal modes and of the single-site entanglement entropy in two quantum quenches that cross phase boundaries of the model and show that these quantities are thermal in the steady state. For our numerical calculations, we use the density matrix renormalization group method for ground-state calculations and time evolution in a Krylov subspace for the quench dynamics as well as exact diagonalization.

Toeplitz operators on the pluriharmonic Hardy space

Corresponding to harmonic Bergman space, in this paper we study Toeplitz operators of pluriharmonic Hardy space. We find the properties of Toeplitz operators on h 2 ( D 2 ) are greatly different from the classical Hardy space, Bergman space and harmonic Bergman space. For example, two analytic Toeplitz operators maybe not semi-commute or commute. Even if it is semi-commute, neither of both symbols is constant; Even if it is commute, their symbols’ nontrivial linear combination may be not constant. We obtain the necessary and sufficient condition of two analytic Toeplitz operators semi-commute and commute on h 2 ( D 2 ).

Wronskians and deep zeros of holomorphic functions

Given linearly independent holomorphic functions f0,…,fn on a planar domain Ω, let E be the set of those points z∈Ω where a nontrivial linear combination ∑j=0nλjfj may have a zero of multiplicity greater than n, once the coefficients λj=λj(z) are chosen appropriately. An elementary argument involving the Wronskian W of the fjʼs shows that E is a discrete subset of Ω (and is actually the zero set of W); thus “deep” zeros are rare. We elaborate on this by studying similar phenomena in various function spaces on the unit disk, with more sophisticated boundary smallness conditions playing the role of deep zeros.

Strongly central sets and sets of polynomial returns mod 1

Central sets in N were introduced by Furstenberg and are known to have substantial combinatorial structure. For example, any central set con- tains arbitrarily long arithmetic progressions, all finite sums of distinct terms of an infinite sequence, and solutions to all partition regular systems of ho- mogeneous linear equations. We introduce here the notions of strongly central and very strongly central, which is as the names suggest are strictly stronger than the notion of central. They are also strictly stronger than syndetic, which in the case of N means that gaps are bounded. Given x 2 R, let w(x) = x b x + 1 c. Kronecker's Theorem says that if 1, 1, 2,..., v are linearly independent over Q and U is a nonempty open subset of ( 1 , 1 ) v , then {x 2 N : (w( 1x),...,w( vx)) 2 U} is nonempty and Weyl showed that this set has positive density. We show here that if 0 is in the closure of U, then this set is strongly central. More generally, let P1,P2,...,Pv be real polynomials with zero constant term. We show that {x 2 N : (w(P1(x)),...,w(Pv(x))) 2 U} is nonempty for every open U with 0 2 c'U if and only if it is very strongly central for every such U and we show that these conclusions hold if and only if any nontrivial rational linear combination of P1,P2,...,Pv has at least one irrational coecient.

The Darboux coordinates for a new family of Hamiltonian operators and linearization of associated evolution equations

A de Sole, V G Kac and M Wakimoto have recently introduced a new family of compatible Hamiltonian operators of the form H(N,0) = D2 ∘ ((1/u) ∘ D)2n ∘ D, where N = 2n + 3, n = 0, 1, 2, …, u is the dependent variable and D is the total derivative with respect to the independent variable. We present a differential substitution that reduces any linear combination of these operators to an operator with constant coefficients and linearizes any evolution equation which is bi-Hamiltonian with respect to a pair of any nontrivial linear combinations of the operators H(N,0). We also give the Darboux coordinates for H(N,0) for any odd N ⩾ 3.

Bounding the number of zeros of certain Abelian integrals

In this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n Abelian integrals to have at most n + k − 1 zeros counted with multiplicities. This condition involves the functions in the integrand of the Abelian integrals and it can be checked, in many cases, in a purely algebraic way.

Asymptotic Formulae for Pairs of Diagonal Cubic Equations

Abstract We investigate the number of integral solutions possessed by a pair of diagonal cubic equations in a large box. Provided that the number of variables in the system is at least fourteen, and in addition the number of variables in any non-trivial linear combination of the underlying forms is at least eight, we obtain an asymptotic formula for the number of integral solutions consistent with the product of local densities associated with the system.

Polynomial algebra for Birkhoff interpolants

We introduce a unifying formulation of a number of related problems which can all be solved using a contour integral formula. Each of these problems requires finding a non-trivial linear combination of possibly some of the values of a function f, and possibly some of its derivatives, at a number of data points. This linear combination is required to have zero value when f is a polynomial of up to a specific degree p. Examples of this type of problem include Lagrange, Hermite and Hermite---Birkhoff interpolation; fixed-denominator rational interpolation; and various numerical quadrature and differentiation formulae. Other applications include the estimation of missing data and root-finding.

Restriction of characters and products of characters

Let G be a finite p-group, for some prime p, and ψ, θ ∈ Irr(G) be irreducible complex characters of G. It has been proved that if, in addition, ψ and θ are faithful characters, then the product ψθ is a multiple of an irreducible or it is the nontrivial linear combination of at least (p + 1)/2 distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k > 0, we can always find a p-group P and irreducible characters Ψ and Θ of P such that the product ΨΘ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.

Pattern sequences in ‹ q, r›- numeration systems

Let q ≥ 2 and 0 ≤ r ≤ q − 2 be integers. In this paper, we study pattern sequences for patterns in ‹ q, r›-numeration systems through their generating functions. Our result implies that any nontrivial linear combination over ℂ of pattern sequences chosen from different ‹ q, r›-numeration systems cannot be a linear recurrence sequence. In particular, pattern sequences in different ‹ q, r›-numeration systems are linearly independent over ℂ, while within one ‹ q, r›-numeration system they can be linearly dependent ℂ.

MinimalE6supersymmetric standard model

We propose a Minimal E6 Supersymmetric Standard Model (ME6SSM) which allows Planck scale unification, provides a solution to the ? problem and predicts a new Z'. Above the conventional GUT scale MGUT ~ 1016 GeV the gauge group corresponds to a left-right symmetric Supersymmetric Pati-Salam model, together with an additional U(1)? gauge group arising from an E6 gauge group broken near the Planck scale. Below MGUT the ME6SSM contains three reducible 27 representations of the Standard Model gauge group together with an additional U(1)X gauge group, consisting of a novel and non-trivial linear combination of U(1)? and two Pati-Salam generators, which is broken at the TeV scale by the same singlet which also generates the effective ? term, resulting in a new low energy Z' gauge boson. We discuss the phenomenology of the new Z' gauge boson in some detail.

Pseudo 1-homogeneous distance-regular graphs

Let Γ be a distance-regular graph of diameter d≥2 and a 1≠0. Let θ be a real number. A pseudo cosine sequence for θ is a sequence of real numbers σ 0,…,σ d such that σ 0=1 and c i σ i−1+a i σ i +b i σ i+1=θ σ i for all i∈{0,…,d−1}. Furthermore, a pseudo primitive idempotent for θ is E θ =s ∑ d i=0 σ i A i , where s is any nonzero scalar. Let \(\hat{v}\) be the characteristic vector of a vertex v∈VΓ. For an edge xy of Γ and the characteristic vector w of the set of common neighbours of x and y, we say that the edge xy is tight with respect to θ whenever θ≠k and a nontrivial linear combination of vectors \(E\hat{x}\) , \(E\hat{y}\) and Ew is contained in \(\mathrm{Span}\{\hat{z}\mid z\in V{\Gamma},\ \partial(z,x)=d=\partial(z,y)\}\) . When an edge of Γ is tight with respect to two distinct real numbers, a parameterization with d+1 parameters of the members of the intersection array of Γ is given (using the pseudo cosines σ 1,…,σ d , and an auxiliary parameter ε). Let S be the set of all the vertices of Γ that are not at distance d from both vertices x and y that are adjacent. The graph Γ is pseudo 1-homogeneous with respect to xy whenever the distance partition of S corresponding to the distances from x and y is equitable in the subgraph induced on S. We show Γ is pseudo 1-homogeneous with respect to the edge xy if and only if the edge xy is tight with respect to two distinct real numbers. Finally, let us fix a vertex x of Γ. Then the graph Γ is pseudo 1-homogeneous with respect to any edge xy, and the local graph of x is connected if and only if there is the above parameterization with d+1 parameters σ 1,…,σ d ,ε and the local graph of x is strongly regular with nontrivial eigenvalues a 1 σ/(1+σ) and (σ 2−1)/(σ−σ 2).