integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

non uniform dependence on initial data of solutions to the euler equations of hydrodynamics we show that continuous dependence on initial data of solutions to the euler equations of incompressible hydrodynamics is optimal more precisely we prove that the data to solution map is not uniformly continuous in sobolev h s topology for any s in mathbb r if the domain is the flat torus mathbb t n mathbb r n 2 pi mathbb z n and for any s 0 if the domain is the whole space mathbb r n

blow up phenomenon for the integrable novikov equation abstract in this paper we investigate a new integrable equation derived recently by v s novikov generalizations of the camassa holm equation j phys a 42 34 2009 342002 14 pp analogous to the camassa holm equation and the degasperis procesi equation this new equation possesses the blow up phenomenon under the special structure of this equation we establish sufficient conditions on the initial data to guarantee the formulation of singularities in finite time a global existence result is also found

polynomial numerical hulls of the direct sum of a normal matrix and a jordan block abstract let j k be the k k jordan block with eigenvalue and let n be an m m normal matrix in this paper we study the polynomial numerical hulls of order 2 and n 1 for a j k n where n m k we obtain a necessary and sufficient condition such that v 2 a has an interior point also we analytically characterize v 2 j 2 n and we show that if n is co linear then v 2 j 2 n a n v 2 j 2 a finally we study v n 1 a and we show that if n is neither co linear nor co circular then v n 1 a has at most one point more than a

global weak solutions for the novikov equation in the paper we mainly study the existence and uniqueness of global weak solutions for the novikov equation we first recall some results and definitions on strong solutions and weak solutions for the equation then we show that the equation has smooth solutions which exist globally in time provided the initial data satisfy certain sign conditions finally we prove the existence and uniqueness of global weak solutions to the equation with the initial data satisfying certain sign conditions

random forests and kernel methods random forests are ensemble methods which grow trees as base learners and combine their predictions by averaging random forests are known for their good practical performance particularly in high dimensional settings on the theoretical side several studies highlight the potentially fruitful connection between the random forests and the kernel methods in this paper we work out this connection in detail in particular we show that by slightly modifying their definition random forests can be rewritten as kernel methods called kerf for kernel based on random forests which are more interpretable and easier to analyze explicit expressions of kerf estimates for some specific random forest models are given together with upper bounds on their rate of consistency we also show empirically that the kerf estimates compare favourably to the random forest estimates

generalizations of the field of values useful in the study of polynomial functions of a matrix for a given square matrix a and positive integer k we consider sets in the complex plane satisfying p a maxz p z for all polynomials p of degree k or less the largest such set referred to as the polynomial numerical hull of degree k was introduced by o nevanlinna convergence of iterations for linear equations birkhauser basel 1993 and a number of properties of this set were derived for both matrices and linear operators we give several equivalent characterizations of the polynomial numerical hull of degree k and we actually compute these sets for several matrices for k 1 this set is just the field of values of a and for k m where m is the degree of the minimal polynomial of a it is the spectrum of a for 1 k m these sets are intermediate between the field of values and the spectrum and sometimes resemble pseudospectra

polynomial numerical hulls of some normal matrices abstract in this note polynomial numerical hulls of matrices of the form a 1 i a 2 where a 1 and a 2 are hermitian are characterized

polynomial numerical hulls of matrices abstract for any n by n complex matrix a we use the joint numerical range w a a 2 a k to study the polynomial numerical hull of order k of a denoted by v k a we give an analytic description of v 2 a when a is normal the result is then used to characterize those normal matrices a satisfying v 2 a a and to show that a unitary matrix a satisfies v 2 a a if and only if its eigenvalues lie in a semicircle where a denotes the spectrum of a when a diag 1 w w n 1 with w e i 2 n we determine v k a for k 2 j n j n 2 we also consider matrices a m n such that a 2 is hermitian for such matrices we show that v 4 a is the spectrum of a and give a description of the set v 2 a

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

polynomial numerical hulls of the direct sum of two jordan blocks abstract let j n be the n n jordan block with a positive real eigenvalue and a j n j n in this paper we study the polynomial numerical hull of degree k h k a when k 2 and k 2 n 1 if 0 3 2 we show that r 2 n r 2 n h 2 a where r 2 n represents the radius of the circular disk h 2 j n 0 and if 2 then there exists 0 such that h 2 a x i y c x 2 y 2 also we give explicit formulas for computing the sets h 3 a and h 2 a r when n 2 although the spectrum and numerical range are continuous set valued functions with respect to the hausdorff metric by an example we show that polynomial numerical hulls may not be even lower semicontinuous

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

well posedness and persistence properties for the novikov equation abstract recently novikov found a new integrable equation we call it the novikov equation in this paper which has nonlinear terms that are cubic rather than quadratic and admits peaked soliton solutions peakons firstly we prove that the cauchy problem for the novikov equation is locally well posed in the besov spaces b 2 r s which generalize the sobolev spaces h s with the critical index s 3 2 then well posedness in h s with s 3 2 is also established by applying kato s semigroup theory finally we present two results on the persistence properties of the strong solution for the novikov equation

complete stagnation of gmres for normal matrices in this paper we study the problem of complete stagnation of the generalized minimum residual gmres method for normal matrices we first characterize all n i n nonsingular normal matrices a such that gmres a b stagnates completely for some vector b also we give necessary and sufficient conditions for the non existence of a real stagnation vector for real normal matrices the number of real stagnation vectors for normal matrices is studied moreover we characterize all the eigenvalues of nonsingular normal matrices a m 3 c such that gmres a b stagnates completely for some b c 3 using the results derived by a greenbaum v ptak and z strakos in 1996 we consider the complete stagnation of unitary matrices and derive another characterization for all nonsingular normal matrices a m 3 r such that gmres a b stagnates completely for some vector b r 3

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

on polynomial numerical hulls of normal matrices abstract the notion of polynomial numerical hull was introduced by o nevanlinna convergence of iteration for linear equations birkhauser 1993 in this note we determine the polynomial numerical hulls of matrices of the form a a 1 i a 2 where a 1 a 2 are hermitian matrices also we study the relationship between rectangular hyperbolas and polynomial numerical hulls of order two for normal matrices the polynomial numerical hulls of order two for some special matrices is studied

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

on the inverse spectral problem for the camassa holm equation abstract a key basis for seeking periodic solutions of the camassa holm equation is to understand the associated spectral problem y 1 4 y my the periodic spectrum can be recovered from the norming constants and the elements of the auxiliary spectrum the potential can then be reconstructed from the periodic spectrum a necessary and sufficient condition for exponential decrease of the widths 2n 2n 1for a sequence 0 m c1 r

on the cauchy problem for the periodic b family of equations and of the non uniform continuity of degasperis procesi equation abstract we consider the problem of the existence of the global solutions and formation of singularities for a b family of equations which includes the camassa holm and degasperis procesi equation we also consider the problem of the uniformly continuity of degasperis procesi equation