In this paper we extend the definition of slice-torus invariant to links. We prove a few properties of the newly-defined slice-torus link invariants: the behaviour under crossing change, a slice genus bound, an obstruction to strong sliceness, and a combinatorial bound. Furthermore, we provide an application to the computation of the splitting number. Finally, we use the slice-torus link invariants and the Whitehead doubling to define new strong concordance invariants for links, which are proven to be independent of the corresponding slice-torus link invariant.

Recently the theory of cardinal polynomial spline interpolation was extended to cardinals -splines [3]. Let

be a polynomial with only real zeros. Denote the set of zeros by . If is the associated differential operator, the null-space

Due to work of $\text{W}$. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on ${{\mathbb{H}}^{3}}$ is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in ${{\text{H}}^{2}}$ where $\text{E}$. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

For any ring we shall denote by the ring of all n × n matrices with elements from and by the set of all polynomials in x with coefficients from .

will denote the non-commutative four-dimensional division algebra of real quaternions with 1, i1, i2, i3 as generators

G. Polya [4] has posed the problem as to whether there are entire transcendental functions of order zero satisfying an algebraic differential equation with rational coefficients. G. Polya himself showed that this is impossible for a first order algebraic differential equation. The general problem is now completely solved. G. Valiron demonstrated an example of a third order algebraic differential equation with an entire transcendental solution of order zero (Theorem 1); V. V. Zimogljad (Theorem 2) proved that every entire transcendental solution of a second order algebraic differential equation is of a positive order. It seems to us expedient to bring these results all together. We give here a proof of Theorem 2 different from and in our view simpler than that of V. V. Zimogljad. Theorem 3 refines the results of G. Polya (and of others, see for example [10]) and establishes an exact lower bound for the order of an arbitrary entire transcendental solution satisfying a first order algebraic differential equation.

Let G be a finite π-separable group, where π is a set of primes. The π-partial characters of G are the restrictions of the ordinary characters to the set of π-elements of G. Such an object is said to be irreducible if it is not the sum of two nonzero partial characters and the set of irreducible π- partial characters of G is denoted Iπ(G). (If p is a prime and π = p′, then Iπ(G) is exactly the set of irreducible Brauer characters at p.)

From their definition, it is obvious that each partial character φ ∊ Iπ(G) can be “lifted” to an ordinary character χ ∊ Irr(G). (This means that φ is the restriction of χ to the π-elements of G.) In fact, there is a known set of canonical lifts Bπ(G) ⊆ Irr(G) for the irreducible π-partial characters. In this paper, it is proved that if 2 ∉ π, then there is an alternative set of canonical lifts (denoted Dπ(G)) that behaves better with respect to character induction.

An application of this theory to M-groups is presented. If G is an M-group and S ⊆ G is a subnormal subgroup, consider a primitive character θ ⊆ Irr(S). It was known previously that if |G : S| is odd, then θ must be linear. It is proved here without restriction on the index of S that θ(1) is a power of 2.

In the early 70's Dennis Sullivan applied his theory of minimal models and surgery to the classification of 1-connected closed smooth manifolds of dimension ≥ 5 up to finite ambiguity [Su]. To a diffeomorphism class of such a manifold M he assigns the isomorphism class given by the real minimal model ℳ (M), the integral structure in form of various lattices and the real Pontryagin classes. If one controls the torsion of the manifolds by some bound, his result is that the map given by the triple above is finite-to-one ([Su], Theorem 13.1). He also proves a realization result for the rational minimal model and the Pontryagin classes but not for the lattices ([Su], Theorem 13.2).

Let A denote a complex (or real) Fréchet algebra (i.e. a complete metrizable locally m-convex algebra, see [2] or [3]). It is known [2] that the topology of such an algebra can be defined by an increasing sequence [qn] (i.e. qn(x) ≦ qn+i(x) for all x £ A and n ≧ 1) of submultiplicative (i.e. qn(xy) ≦ qn(x)qn(y) for all x, 3/ Ç ^4 and for each n ≧ I) seminorms.

The approximately continuous integral which includes the Lebesgue integral has been considered by Burkill [1] and Ridder [3; 4]. I [2] also defined the AD-integral of this kind which is more general than the AP-integral of Burkill [1] and the Denjoy integral in the wide sense. But this integral is equivalent to the β-integral of Ridder.

Our aim in this paper is to characterize the AD-integral in the following way: The AD-integral is the least general approximately continuous integral (Definition 1) which includes the Lebesgue integral and fulfils the Cauchy and Harnack conditions (Definition 2).

In an earlier paper [7], we have studied the existence, uniqueness and asymptotic behavior of solutions to certain infinite systems of linear differential equations with constant coefficients. In the present paper we are interested in systems of nonlinear equations whose coefficients are not necessarily constants; more specifically, we are concerned with infinite systems of the form

“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius $\left( \text{QcF} \right)$ coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is $\text{QcF}$ if it is weakly isomorphic to its (left, or right) rational dual $\text{Rat}\left( {{C}^{*}} \right)$ in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, $\text{QcF}$, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra $A$ that is isomorphic to its complete topological dual ${{A}^{\vee }}$. We show that $A$ is a (quasi)Frobenius algebra if and only if $A$ is the dual ${{C}^{*}}$ of a (quasi)co-Frobenius coalgebra $C$. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer $q$-homological algebra, topology or graph theory, showing the importance of the concept.

Let $E/\mathbb{Q}$ be an elliptic curve defined by the equation ${{y}^{2}}\,=\,{{x}^{3}}\,+\,ax\,+\,b.$ For a prime $p$, $p\nmid \Delta \,=\,-16\left( 4{{a}^{3}}+27{{b}^{2}} \right)\,\ne \,0,$ define

$${{N}_{p}}=p+1-{{a}_{p}}=\,\left| E\left( {{\mathbb{F}}_{p}} \right) \right|.$$

As a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant $c$,

$$\prod\limits_{p\le x,p\nmid \Delta }{\frac{{{N}_{p}}}{p}\,\sim \,c{{\left( \log x \right)}^{r}},\,\,\,x\to \infty .}$$

Let ${{\alpha }_{p}}$ and ${{\beta }_{p}}$ be the eigenvalues of the Frobenius at $p$. Define

$${{\tilde{c}}_{n}}=\left\{ \begin{align} & \frac{\alpha _{p}^{k}+\beta _{P}^{k}}{k}\,\,\,\,n={{p}^{k}},\,p\,\text{is}\,\text{a}\,\text{prime,}\,k\,\text{is}\,\text{a}\,\text{natural}\,\text{number},\,p\nmid \Delta . \\ & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \\ \end{align} \right.\,$$

and $\tilde{C}\left( x \right)\,=\,\sum\nolimits_{n\le x}{{{{\tilde{c}}}_{n}}.}$ In this paper, we establish the equivalence between the conjecture and the condition $\tilde{C}\left( x \right)\,=\,\mathbf{o}\left( x \right).$ The asymptotic condition is indeed much deeper than what we know so far or what we can know under the analogue of the Riemann hypothesis. In addition, we provide an oscillation theorem and an $\Omega $ theorem which relate to the constant $c$ in the conjecture.

By “algebra” we shall mean a finitary universal algebra, that is, a pair 〈A; F〉 where A and F are nonvoid sets and every element of F is a function, defined on A, of some finite number of variables. Armbrust and Schmidt showed in [1] that for any finite nonvoid set A, every group G of permutations of A is the automorphism group of an algebra defined on A and having only one operation, whose rank is the cardinality of A. In [6], Jónsson gave a necessary and sufficient condition for a given permutation group to be the automorphism group of an algebra, whereupon Plonka [8] modified Jonsson's condition to characterize the automorphism groups of algebras whose operations have ranks not exceeding a prescribed bound.

We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

It is well known that the length Zn of the longest head run observed in n tosses with a fair coin is approximately equal to log2n with a stochastically bounded remainder term. Though — log2n does not converge in law, in the present paper it is shown to have almost sure limit distribution in the sense of the a. s. central limit theorem having been studied recently. The results are formulated and proved in a general setup covering other interesting problems connected with patterns and runs such as the longest monotone block or the longest tube of a random walk.

In the study of approximate methods for solving ordinary differential equations, an interesting question arises. To state it roughly for a single first order expression, let y0(t) be the solution of the equation

(1.1)

which satisfies the initial condition y(a) = na. Let nb be an approximation to the value of y0 at a later time, t = b.

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form

$${\nabla }'P\left( x \right)\nabla u+\text{HR}u+\text{{S}'G}u\text{+ F}u\text{=}\text{f}\,\text{+}\,\text{{T}'g}\,\text{in}\,\Theta$$

$$u=\varphi \,\,on\,\,\partial \Theta$$

The principal part ${\xi }'P\left( x \right)\xi$ of the above equation is assumed to be comparable to a quadratic form $\mathcal{Q}\left( x,\xi \right)={\xi }'Q\left( x \right)\xi$ that may vanish for non-zero $\xi \in {{\mathbb{R}}^{n}}$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $Q{{H}^{1}}\left( \Theta \right)={{W}^{1,2}}\left( \Theta ,Q \right)$ and $QH_{0}^{1}\left( \Theta \right)=W_{0}^{1,2}\left( \Theta ,Q \right)$ as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

We consider the $p$-Yang-Mills functional $\left( p\,\ge \,2 \right)$ defined as $Y{{M}_{p}}(\nabla ):=\frac{1}{p}{{\int }_{M}}{{\left\| {{R}^{\nabla }} \right\|}^{p}}$. We call critical points of $Y{{M}_{p}}(\cdot )$ the p-Yang–Mills connections, and the associated curvature ${{R}^{\nabla }}$ the $p$-Yang-Mills fields. In this paper, we prove gap properties and instability theorems for $p$-Yang-Mills fields over submanifolds in ${{\mathbb{R}}^{n+k}}$ and ${{\mathbb{S}}^{n+k}}$.

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that

Let $G$ be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations $(T,\mathcal{U})$ of $G$ on a Banach space $\mathcal{U}$ by giving a $G$-orbit in ${{n}^{*}}$ ($\mathfrak{n}$ being the nilradical of $\mathfrak{g}$), a topologically irreducible representation of ${{L}^{1}}({{\mathbb{R}}^{n}},\,\,\omega )$ , for a certain weight $\omega $ and a certain $n\,\in \,\mathbb{N}$, and a topologically simple extension norm. If $G$ is not symmetric, i.e., if the weight $\omega $ is exponential, we get a new type of representations which are fundamentally different from the induced representations.