Vanishing Of Sum Research Articles

On Weakly Cyclic Generalized Z-Symmetric Manifolds

In this paper, we define a (0,2)-type symmetric tensor Z and called it a generalized Z tensor. We study weakly cyclic generalized Z-symmetric manifold and prove that it is a generalized quasi-Einstein manifold. We have obtained a condition for vanishing of sum of 1-forms. We further find a condition to be a Ricci semisymmetric manifold. Finally, it is shown that the semisymmetry and Weyl semisymmetry are equivalent in such a manifold.

Balanced Factorizations

Any rational number can be factored into a product of several rationals whose sum vanishes. This simple but nontrivial fact was suggested as a problem on a mathematical olympiad for high school students. We completely solve similar questions in all finite fields and in some other rings, e.g., in the complex and real matrix algebras. Also, we state several open questions.

Roles of antinucleon degrees of freedom in the relativistic random phase approximation

Roles of antinucleon degrees of freedom in the relativistic random phase approximation(RPA) are investigated. The energy-weighted sum of the RPA transition strengths is expressed in terms of the double commutator between the excitation operator and the Hamiltonian, as in nonrelativistic models. The commutator, however, should not be calculated with a usual way in the local field theory, because, otherwise, the sum vanishes. The sum value obtained correctly from the commutator is infinite, owing to the Dirac sea. Most of the previous calculations takes into account only a part of the nucleon-antinucleon states, in order to avoid the divergence problems. As a result, RPA states with negative excitation energy appear, which make the sum value vanish. Moreover, disregarding the divergence changes the sign of nuclear interactions in the RPA equation which describes the coupling of the nucleon particle-hole states with the nucleon-antinucleon states. Indeed, excitation energies of the spurious state and giant monopole states in the no-sea approximation are dominated by those unphysical changes. The baryon current conservation can be described without touching the divergence problems. A schematic model with separable interactions is presented, which makes the structure of the relativistic RPA transparent.

The black hole interior and a curious sum rule

We analyze the Euclidean geometry near non-extremal NS5-branes in string theory, including regions beyond the horizon and beyond the singularity of the black brane. The various regions have an exact description in string theory, in terms of cigar, trumpet and negative level minimal model conformal field theories. We study the worldsheet elliptic genera of these three superconformal theories, and show that their sum vanishes. We speculate on the significance of this curious sum rule for black hole physics.

One loop tests of higher spin AdS/CFT

Vasiliev’s type A higher spin theories in AdS4 have been conjectured to be dual to the U(N) or O(N) singlet sectors in 3-d conformal field theories with N-component scalar fields. We compare the $ \mathcal{O} $ (N 0) correction to the 3-sphere free energy F in the CFTs with corresponding calculations in the higher spin theories. This requires evaluating a regularized sum over one loop vacuum energies of an infinite set of massless higher spin gauge fields in Euclidean AdS4. For the Vasiliev theory including fields of all integer spin and a scalar with Δ = 1 boundary condition, we show that the regularized sum vanishes. This is in perfect agreement with the vanishing of subleading corrections to F in the U(N) singlet sector of the theory of N free complex scalar fields. For the minimal Vasiliev theory including fields of only even spin, the regularized sum remarkably equals the value of F for one free real scalar field. This result may agree with the O(N) singlet sector of the theory of N real scalar fields, provided the coupling constant in the Vasiliev theory is identified as G N ~ 1/(N − 1). Similarly, consideration of the USp(N) singlet sector for N complex scalar fields, which we conjecture to be dual to the husp(2; 0|4) Vasiliev theory, requires G N ~ 1/(N + 1). We also test the higher spin AdS3 /CFT2 conjectures by calculating the regularized sum over one loop vacuum energies of higher spin fields in AdS3. We match the esult with the $ \mathcal{O} $ (N 0) term in the central charge of the W N minimal models; this requires a certain truncation of the CFT operator spectrum so that the bulk theory contains two real scalar fields with the same boundary conditions.

On a transform method for the Laplace equation in a polygon

Let q(x, y) satisfy a boundary value problem for the Laplace equation in an arbitrary convex polygon with n sides. An integral representation in the complex k‐plane is given for q(x, y) in terms of n functions ρj(k), j = 1, …, n. The function ρj consists of an integral over the jth side involving both qx and qy , thus each ρj involves one unknown boundary value. The functions ρj are not independent but they satisfy the important global relation that their sum vanishes. The solution of a given boundary value problem reduces to the analysis of this single relation for the n unknown ρj. For a general polygon with general Poincaré boundary conditions, this gives rise to a matrix Riemann–Hilbert problem. In this paper it is shown that for simple polygons and for a large class of boundary conditions, the above Riemann–Hilbert problem (a) can either be reduced to a triangular RH problem which can be solved in closed form or (b) can be bypassed, and the ρj can be obtained using only algebraic manipulations. As an illustration of these ‘triangular’ and ‘algebraic’ cases we solve the Laplace equation in the quarter‐plane, the semi‐infinite strip and the right isosceles triangle with certain Poincaré boundary conditions. These boundary value problems, which include the Dirichlet and the Neumann problems as particular cases, cannot be solved by conformal mappings.

Spontaneous localization of bulk matter fields

We study models compactified on S 1/ Z 2 with bulk and brane matter fields charged under U(1) gauge symmetry. We calculate the FI-terms and show by minimizing the resulting potential that supersymmetry or gauge symmetry is spontaneously broken if the sum of the charges does not vanish. Even if this sum vanishes, there could be an instability as a consequence of localized FI-terms. This leads to a spontaneous localization of charged bulk fields on respective branes.

Asymptotic analysis of error probabilities for the nonzero-mean Gaussian hypothesis testing problem

Using a large-deviation theory approach, the rate at which the probability of detection error vanishes as sample size increases in the testing of nonzero-mean Gaussian stochastic processes is studied. After suitable transformation, the likelihood ratio test statistic is expressed as a sum of independent Gaussian random variables. The precise asymptotic rate at which the tail probability of this sum vanishes is derived by use of Ellis' theorem in conjunction with asymptotic analysis of Toeplitz matrices. As a specific example, a signal composed of a deterministic mean component, a zero-mean stochastic component, and a white-noise background was tested against white noise alone. Results confirm the obvious: for fixed stochastic signal power the rate of error decrease increases as the power in the deterministic mean increases. With higher signal-to-noise values, the probability of error must vanish more quickly. For fixed deterministic mean component, as the stochastic signal power increases there is curious dip in the rate of error decrease; however, as this power is increased, eventually the rate of error decrease increases.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Many-body interactions in physisorption: Nonadditivity of film and substrate contributions to the holding potential

Calculations are presented of non-pair-additive contributions to the interaction energy of a probe atom with a physisorbed monolayer film and substrate. The Axilrod-Teller-Muto triple dipole energy is summed for triplets composed of probe atom, monolayer atom, and substrate atom. The lateral average, over probe atom positions, of the sum vanishes. A similar result and a nonvanishing term from a higher many-body interaction are derived also from the McLachlan theory of the dispersion interaction of two atoms in the presence of a continuum substrate.

Finite-temperature contributions to the magnetic susceptibility of a normal Fermi liquid

We show that long-wavelength spin and density fluctuations give no ${T}^{2}\mathrm{ln}T$ contribution to the magnetic susceptibility of a normal Fermi liquid. The calculations are made within the framework of Landau Fermi-liquid theory. We find that while there are ${T}^{2}\mathrm{ln}T$ terms arising from both the quasiparticle density of states and from the Landau quasiparticle interaction function, their sum vanishes. In calculating the Landau quasiparticle interaction, it is found to be important to include processes involving two long-wavelength fluctuations, in addition to the exchange of a single fluctuation.

Second-Order Corrections to Triangle Anomalies in an Abelian Gauge Theory

We consider an Abelian gauge theory of weak interactions which is arranged so that in lowest order there are no anomalies in the divergence of the axial-vector current. This is done by having two fermions whose couplings to the axial-vector current are equal in magnitude but opposite in sign. We then show that while the second-order corrections to the anomaly are nonzero for each of the fermions, their sum vanishes, ensuring that the theory is anomaly-free through second order.

A Vanishing Finite Sum associated with Jacobi's triple product identity

Jacobi's triple product identity is shown to be equivalent to the vanishing of a finite sum. The vanishing of the sum is then established independently of Jacobi's identity, and the behavior of a related finite sum is discussed. Finally, the possibility of generalizing Jacobi's identity to include variables with non-quadratic exponents is examined; no decisive result is obtained.

A note on the cyclotomic polynomial

The n-th roots of unity 1, ω, …, ωn-1, where ω = exp (2πi\n), are linearly dependent in the field Q of rationals since, for instance, their sum vanishes. We are here concerned with the linear relations between them with integral coefficients. Let U denote the vector space of elements u = (u0, …, un−1) over Q and let N be the subspace of elements u defined by the relation u0+u1ω+…+un−1ωn−1=0. (1)