Graded Symmetry Algebras of Time-Dependent Evolution Equations and Application to the Modified KP equations

By starting from known graded Lie algebras, including Virasoro algebras, new kinds of time-dependent evolution equations are found possessing graded symmetry algebras. The modified KP equations are taken as an illustrative example: new modified KP equations with $m$ arbitrary time-dependent coefficients are obtained possessing symmetries involving $m$ arbitrary functions of time. A particular graded symmetry algebra for the modified KP equations is derived in this connection homomorphic to the Virasoro algebras.Comment: 19 pages, latex, to appear in J. Nonlinear Math. Phy

The Knowledge Graph for End-to-End Learning on Heterogeneous Knowledge

In modern machine learning,raw data is the preferred input for our models. Where a decade ago data scientists were still engineering features, manually picking out the details we thought salient, they now prefer the data in their raw form. As long as we can assume that all relevant and irrelevant information is present in the input data, we can design deep models that build up intermediate representations to sift out relevant features. However, these models are often domain specific and tailored to the task at hand, and therefore unsuited for learning on heterogeneous knowledge: information of different types and from different domains. If we can develop methods that operate on this form of knowledge, we can dispense with a great deal more ad-hoc feature engineering and train deep models end-to-end in many more domains. To accomplish this, we first need a data model capable of expressing heterogeneous knowledge naturally in various domains, in as usable a form as possible, and satisfying as many use cases as possible. We argue that the knowledge graph is a suitable candidate for this data model. We further discuss some of the promises and challenges of this approach, and how we are currently broadening our efforts to multi-modal knowledge graphs

Funeral Services for Rev. F. D. Hemenway

The sermon is in printed pamphlet form.Funeral services of Rev. F. D. Hemenway by Rev. W. X. Ninde. Presented April 22, 1884 at Garrett Biblical Institute

A generalized mixed type of quartic, cubic, quadratic and additive functional equation

We determine the general solution of the functional equation f(x+ky)+f(x−ky) = g(x+y)+g(x−y)+ +h(x)+h˜(y) for fixed integers k with k 6= 0, ±1 without assuming any regularity condition on the unknown functions f, g, h, h˜. The method used for solving these functional equations is elementary but exploits an important result due to Hosszu. The solution of this functional equation can also be determined in certain type ´ of groups using two important results due to SzekelyhidiВизначено загальний розв’язок функцiонального рiвняння f(x + ky) + f(x − ky) = g(x + y) + + g(x − y) + h(x) + h˜(y) для фiксованих цiлих k при k 6= 0, ±1 без припущення наявностi будь-якої умови регулярностi для невiдомих функцiй f, g, h, h˜. Метод, що використано для розв’язку цих функцiональних рiвнянь, елементарний, але базується на важливому результатi Хозу. Розв’язок цього функцiонального рiвняння може бути визначений у певному типi груп з використанням двох важливих результатiв Чекелiхiдi

On integrability of a (2+1)-dimensional perturbed Kdv equation

A (2+1)-dimensional perturbed KdV equation, recently introduced by W.X. Ma and B. Fuchssteiner, is proven to pass the Painlev\'e test for integrability well, and its 4$\times$4 Lax pair with two spectral parameters is found. The results show that the Painlev\'e classification of coupled KdV equations by A. Karasu should be revised

The Knowledge Graph for End-to-End Learning on Heterogeneous Knowledge

In modern machine learning,raw data is the preferred input for our models. Where a decade ago data scientists were still engineering features, manually picking out the details we thought salient, they now prefer the data in their raw form. As long as we can assume that all relevant and irrelevant information is present in the input data, we can design deep models that build up intermediate representations to sift out relevant features. However, these models are often domain specific and tailored to the task at hand, and therefore unsuited for learning on heterogeneous knowledge: information of different types and from different domains. If we can develop methods that operate on this form of knowledge, we can dispense with a great deal more ad-hoc feature engineering and train deep models end-to-end in many more domains. To accomplish this, we first need a data model capable of expressing heterogeneous knowledge naturally in various domains, in as usable a form as possible, and satisfying as many use cases as possible. We argue that the knowledge graph is a suitable candidate for this data model. We further discuss some of the promises and challenges of this approach, and how we are currently broadening our efforts to multi-modal knowledge graphs