Explicit Integral Formula Research Articles

The Riemann zeta function and exact exponential sum identities of divisor functions

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities involving the divisor functions, and we provide examples of these identities. We conjecturally propose a method to compute divisor functions by matrix inversion, without employing arithmetic techniques.

Exploring Explicit Definite Integral Formulae with Trigonometric and Hyperbolic Functions

Making use of integration by parts and variable replacement methods, we derive some interesting explicit definite integral formulae involving trigonometric or hyperbolic functions, whose results are expressed in terms of Catalan’s constant, Dirichlet’s beta function, and Riemann’s zeta function, as well as π in the denominator.

Borel (α,β)-multitransforms and quantum Leray–Hirsch: Integral representations of solutions of quantum differential equations for [formula omitted]-bundles

In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (qDEs) associated with the quantum cohomology of P1-bundles on Fano varieties. It is shown that bases of solutions of the qDE associated with the total space of the P1-bundle can be reconstructed from the datum of bases of solutions of the qDE associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in [17], called Borel(α,β)-multitransforms. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a universal feature, being independent of the specifically chosen P1-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of qDEs. As an example, we show how to integrate the qDE of blow-up of P2 at one point via Borel multitransforms of solutions of the qDE of P1.

On a semilinear wave equation in anti-de Sitter spacetime: The critical case

In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.

Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices

In [I. Dumitriu and E. Paquette, Spectra of overlapping Wishart matrices and the gaussian free field, Random Matrices: Theory Appl. 07(2) (2018) 1850003], the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic evolution, the fluctuations asymptotically converge to a three-dimensional Gaussian field, which has an explicit contour integral formula. This is analogous to the result of [A. Borodin, CLT for spectra of submatrices of Wigner random matrices, Moscow Math. J. 14(1) (2014) 29–38] for stochastic Wigner matrices.

Emergent Sasaki-Einstein geometry and AdS/CFT

A central problem in any quantum theory of gravity is to explain the emergence of the classical spacetime geometry in some limit of a more fundamental, microscopic description of nature. The gauge/gravity-correspondence provides a framework in which this problem can, in principle, be addressed. This is a holographic correspondence which relates a supergravity theory in five-dimensional Anti-deSitter space to a strongly coupled superconformal gauge theory on its 4-dimensional flat Minkowski boundary. In particular, the classical geometry should therefore emerge from some quantum state of the dual gauge theory. Here we confirm this by showing how the classical metric emerges from a canonical state in the dual gauge theory. In particular, we obtain approximations to the Sasaki-Einstein metric underlying the supergravity geometry, in terms of an explicit integral formula involving the canonical quantum state in question. In the special case of toric quiver gauge theories we show that our results can be computationally simplified through a process of tropicalization.

High order Melnikov method: Pendulums

In this paper, we present an explicit integral formula for the high order Melnikov function D1(t0) for periodically perturbed pendulum equations. The acquired integral formula is then applied to a specific pendulum equation to offer an example of high frequency perturbation, of which the classical Melnikov function D0(t0) fails to dominate the higher order term εD1(t0).

Two New Generalizations of Extended Bernoulli Polynomials and Numbers, and Umbral Calculus

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

Learning modified indicator functions for surface reconstruction

Surface reconstruction is a fundamental problem in 3D graphics. In this paper, we propose a learning-based approach for implicit surface reconstruction from raw point clouds without normals. Our method is inspired by Gauss Lemma in potential energy theory, which gives an explicit integral formula for the indicator functions. We design a novel deep neural network to perform surface integral and learn the modified indicator functions from un-oriented and noisy point clouds. We concatenate features with different scales for accurate point-wise contributions to the integral. Moreover, we propose a novel Surface Element Feature Extractor to learn local shape properties. Experiments show that our method generates smooth surfaces with high normal consistency from point clouds with different noise scales and achieves state-of-the-art reconstruction performance compared with current data-driven and non-data-driven approaches.

Tractable Relaxations of Composite Functions

In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes [Formula: see text] time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation.

Toeplitz Operators, $$\mathbb {T}^m$$-Invariance and Quasi-homogeneous Symbols

For a partition \(\varvec{k} = (k_1, \dots , k_m)\) of n consider the group \(\mathrm {U}(\varvec{k}) = \mathrm {U}(k_1) \times \dots \times \mathrm {U}(k_m)\) block diagonally embedded in \(\mathrm {U}(n)\) and the center \(\mathbb {T}^m\) of \(\mathrm {U}(\varvec{k})\). We study the Toeplitz operators with \(\mathbb {T}^m\)-invariant symbols acting on the weighted Bergman spaces on the unit ball \(\mathbb {B}^n\). We introduce the \((\varvec{k},j)\)-quasi-radial quasi-homogeneous symbols as those that are invariant under the group \(\mathrm {U}(\varvec{k},j,\mathbb {T})\) obtained from \(\mathrm {U}(\varvec{k})\) by replacing the factor \(\mathrm {U}(k_j)\) with its center \(\mathbb {T}\). These symbols are used to build commutative Banach non-\(C^*\) algebras generated by Toeplitz operators. These algebras generalize those from the literature and show that they can be built using groups. We describe the action of such Toeplitz operators on monomials through explicit integral formulas involving the symbols. We prove that every Toeplitz operator with \(\mathbb {T}^m\)-invariant symbol has an associated Toeplitz operator with \(\mathrm {U}(\varvec{k})\)-invariant symbol in terms of which we can describe some properties.

Refined Cauchy identity for spin Hall–Littlewood symmetric rational functions

Fully inhomogeneous spin Hall–Littlewood symmetric rational functions Fλ arise in the context of sl(2) higher spin six vertex models, and are multiparameter deformations of the classical Hall–Littlewood symmetric polynomials. We obtain a refined Cauchy identity expressing a weighted sum of the product of two Fλ's as a determinant. The determinant is of Izergin–Korepin type: it is the partition function of the six vertex model with suitably decorated domain wall boundary conditions. The proof of equality of two partition functions is based on the Yang–Baxter equation.We rewrite our Izergin–Korepin type determinant in a different form which includes one of the sets of variables in a completely symmetric way. This determinantal identity might be of independent interest, and also allows to directly link the spin Hall–Littlewood rational functions with (the Hall–Littlewood particular case of) the interpolation Macdonald polynomials. In a different direction, a Schur expansion of our Izergin–Korepin type determinant yields a deformation of Schur symmetric polynomials.In the spin-12 specialization, our refined Cauchy identity leads to a summation identity for eigenfunctions of the ASEP (Asymmetric Simple Exclusion Process), a celebrated stochastic interacting particle system in the Kardar–Parisi–Zhang universality class. This produces explicit integral formulas for certain multitime probabilities in ASEP.

Muckenhoupt type weights and Berezin formulas for Bergman spaces

By means of Muckenhoupt type conditions, we characterize the weights ω on C such that the Bergman projection of Fα2,ℓ=H(C)∩L2(C,e−α2|z|2ℓ), α>0, ℓ>1, is bounded on Lp(C,e−αp2|z|2ℓω(z)), for 1<p<∞. We also obtain explicit representation integral formulas for functions in the weighted Bergman spaces Ap(ω)=H(C)∩Lp(ω). Finally, we check the validity of the so called Sarason conjecture about the boundedness of products of certain Toeplitz operators on the spaces Fαp,ℓ=H(C)∩Lp(C,e−αp2|z|2ℓ).

A Review of Matrix SIR Arino Epidemic Models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”,which may be useful to obtain approximate control policies.

A multi-material transport problem with arbitrary marginals

In this paper we study general transportation problems in \({\mathbb {R}}^n\), in which m different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures \(\mu ^-\), \(\mu ^+\) on \({\mathbb {R}}^n\) with values in \({\mathbb {R}}^m\). When the measures are finite atomic, a discrete transportation network is a measure T on \({\mathbb {R}}^n\) with values in \({\mathbb {R}}^{n\times m}\) represented by an oriented graph \({\mathcal {G}}\) in \({\mathbb {R}}^n\) whose edges carry multiplicities in \({\mathbb {R}}^m\). The constraint is encoded in the relation \({\mathrm{div}}(T)=\mu ^- -\mu ^+\). The cost of the discrete transportation T is obtained integrating on \({\mathcal {G}}\) a general function \({\mathcal {C}}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}\) of the multiplicity. When the initial data \(\left( \mu ^-,\mu ^+\right) \) are arbitrary (possibly diffuse) measures, the cost of a transportation network between them is computed by relaxation of the functional on graphs mentioned above. Our main result establishes the existence of cost-minimizing transportation networks for arbitrary data \(\left( \mu ^-,\mu ^+\right) \). Furthermore, under additional assumptions on the cost integrand \({\mathcal {C}}\), we prove the existence of transportation networks with finite cost and the stability of the minimizers with respect to variations of the given data. Finally, we provide an explicit integral representation formula for the cost of rectifiable transportation networks, and we characterize the costs such that every transportation network with finite cost is rectifiable.

A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied.

Twisted Traces and Positive Forms on Quantized Kleinian Singularities of Type A

Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017), 345-392, arXiv:1601.05378] and [Etingof P., Stryker D., SIGMA 16 (2020), 014, 28 pages, arXiv:1909.13588], we undertake a detailed study of twisted traces on quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we give explicit integral formulas for these traces and use them to determine when a trace defines a positive Hermitian form on the corresponding algebra. This leads to a classification of unitary short star-products for such quantizations, a problem posed by Beem, Peelaers and Rastelli in connection with 3-dimensional superconformal field theory. In particular, we confirm their conjecture that for $n\le 4$ a unitary short star-product is unique and compute its parameter as a function of the quantization parameters, giving exact formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$, this, in particular, recovers the theory of unitary spherical Harish-Chandra bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed as a starting point for a generalization of the theory of unitary Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras [Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we derive recurrences to compute the coefficients of short star-products corresponding to twisted traces, which are generalizations of discrete Painlev\'e systems.

A higher moment formula for the Siegel–Veech transform over quotients by Hecke triangle groups

We compute higher moments of the Siegel–Veech transform over quotients of SL (2,\mathbb{R}) by the Hecke triangle groups. After fixing a normalization of the Haar measure on SL (2,\mathbb{R}) we use geometric results and linear algebra to create explicit integration formulas which give information about densities of k -tuples of vectors in discrete subsets of \mathbb{R}^2 which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over SL (2,\mathbb{R})/ SL (2,\mathbb{Z}) .

Truncated-Exponential-Based Appell-Type Changhee Polynomials

The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials Chn*(x) (10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials eCn*(x) in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials eCn*(x). Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested.

Numerical results for Saito’s uniqueness theorem in inverse scattering theory

We consider an inverse scattering problem for the Schrödinger operator in two dimensions. The aim of this work is to discuss some first numerical results on Saito’s formula. Saito’s formula is an explicit integral formula, which at the high-frequency limit gives a uniqueness result for the inverse scattering problem. The numeric approach is quite straight-forward: we take a large enough fixed wave number and evaluate the integrals in Saito’s formula numerically. The potential function can then be recovered from the blurry measurements by using the fast Fourier transform and a high-pass filter. We also discuss in detail how the synthetic data is generated via a matrix-based approach. Several numerical examples are shown to demonstrate the results.