Inverse spectral problems for non-self-adjoint Sturm–Liouville operators with discontinuous boundary conditions

This paper deals with the inverse spectral problem for a non-self-adjoint Sturm–Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential q is known a priori on a subinterval $$ \left[ b,\pi \right] $$ with $$b\in \left( d,\pi \right] $$ or $$b=d$$, then $$h,\,\beta ,\,\gamma $$ and q on $$\left[ 0,\pi \right] $$ can be uniquely determined by partial spectral data consisting of a sequence of eigenvalues and a subsequence of the corresponding generalized normalizing constants or a subsequence of the pairs of eigenvalues and the corresponding generalized ratios. For the case $$b\in \left( 0,d\right) $$, a similar statement holds if $$\beta ,\,\gamma $$ are also known a priori. Moreover, if q satisfies a local smoothness condition, we provide an alternative approach instead of using the high-energy asymptotic expansion of the Weyl m-function to solve the problem of missing eigenvalues and norming constants.     

Remarks on Rawnsley’s $$\varvec{\varepsilon }$$ε -function on the Fock–Bargmann–Hartogs domains

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kähler metric associated with the Kähler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.     

Counter-examples to the Dunford–Schwartz pointwise ergodic theorem on $$\varvec{L^1+L^\infty }$$ L 1 + L ∞

Extending a result by Chilin and Litvinov, we show by construction that given any $$\sigma $$ -finite infinite measure space $$(\Omega ,\mathcal {A}, \mu )$$ and a function $$f\in L^1(\Omega )+L^\infty (\Omega )$$ with $$\mu (\{|f|>\varepsilon \})=\infty $$ for some $$\varepsilon >0$$ , there exists a Dunford–Schwartz operator T over $$(\Omega ,\mathcal {A}, \mu )$$ such that $$\frac{1}{N}\sum _{n=1}^N (T^nf)(x)$$ fails to converge for almost every $$x\in \Omega $$ . In addition, for each operator we construct, the set of functions for which pointwise convergence fails almost everywhere is residual in $$L^1(\Omega )+L^\infty (\Omega )$$ .     

Markoff–Rosenberger triples and generalized Lucas sequences

Periodica Mathematica Hungarica - We consider the Markoff–Rosenberger equation $$\begin{aligned} ax^2+by^2+cz^2=dxyz \end{aligned}$$ with $$(x,y,z)=(U_i,U_j,U_k)$$ , where $$U_i$$ denotes the...     

The equation $$x^4+2^ny^4=z^4$$ x 4 + 2 n y 4 = z 4 in algebraic number fields

Acta Mathematica Hungarica - Let n be a positive integer. We show that if the equation $$(1) \qquad \qquad \qquad x^4+2^ny^4=z^4$$ has a solution (x,y,z) in a cubic number field K with $$xyz \neq...     

The Banach–Mazur game and domain theory

Archiv der Mathematik - We prove that player $$\alpha $$ has a winning strategy in the Banach–Mazur game on a space X if and only if X is F-Y countably $$\pi $$-domain representable. We show...     

A Levi–Civita functional equation on semigroups

Aequationes mathematicae - Let S be a semigroup. We describe the solutions $$f,g:S \rightarrow \mathbb {C}$$ of the functional equation $$\begin{aligned} f(xy) = f(x)g(y) + g(x)f(y) - g(x)g(y), \...     

Cameron–Liebler k-sets in subspaces and non-existence conditions

Designs, Codes and Cryptography - In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron–Liebler line classes in PG $$(n,q), n\ge 3$$ , to...     

Nonlinear elliptic equations of order 2m and subdifferentials

LetA be an operator of the calculus of variations of order 2m onW ^m,p (Ω) andj a normal convex integrand. Forf ∈L ^p (Ω), the equation $$\mathcal{A}u + \partial j(x,u) \ni f, in \Omega , u - \phi \in W_0^{m,p} (\Omega ),$$ may have no strong solutions whenm>1, even ifj is independent ofx and φ=0. However, we obtain existence results whenj is everywhere finite and $$\int_\Omega {j(x,\phi ) dx< + \infty ,} $$ by the study of the subdifferential of the function $$\upsilon \mapsto \int_\Omega {j(x,\upsilon + \phi ) dx on W_0^{m,p} (\Omega ).} $$      

The <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> sphere of a 4-manifold

Let denote the set of even integers . We prove that when H ≥ X ^0.33, almost all integers can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.      

Class number one problem for the real quadratic fields $${{\mathbb {Q}({\sqrt{m^2+2r}})}}$$ Q ( m 2 + 2 r )

Archiv der Mathematik - We investigate the class number one problem for a parametric family of real quadratic fields of the form $$\mathbb {Q}( \sqrt{m^2+4r})$$ for certain positive integers m and r.     

Zur Bestimmung vonE n [x n+2m ]

Remark on the estimation ofE _n [x ^n+2m ]. Let be $$E_n [f]: = \mathop {\inf }\limits_{p \in P_n } \mathop {\sup }\limits_{x \in [ - 1, 1]} |f(x) - p(x)|$$ (P _n : set of all polynomials of degreen). Riess-Johnson [4] proved (3) $$E_n [x^{n + 2m} ] = \frac{{n^{m - 1} }}{{2^{n + 2m - 1} (m - 1)!}}[1 + O(n^{ - 1} )],n even.$$ This degree of approximation is realized by expansion in Chebyshev polynomials and by interpolation at Chebyshev nodes. The purpose of this paper is to give a more precise estimation by constructing the polynomial of best approximation on a finite set. This construction is easily done and one obtains the result, that the termO(n ^?1) in (3) may be replaced by 1/2(m ? 1) (3m + 2)n ^?1 + O(n ^?2).     

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

Tropical Duality in $$(d+2)$$ ( d + 2 ) -Angulated Categories

Applied Categorical Structures - Let $$\mathscr {C}$$ be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories $$\mathscr {T}$$ and $$\mathscr {U}$$ . A result from...     

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Harnack inequality for Ornstein–Uhlenbeck type operators

Archiv der Mathematik - We prove Harnack type inequalities for non-negative weak solutions in $$(0,T]\times \mathbb {R}^N$$ of parabolic problems related to operators of the type $$L=\hbox...     

Distance-Regular Shilla Graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript> = <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript>

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ₁ equal to a₃. For a Shilla graph, let us put a = a₃ and b = k/a. It is proved in this paper that a Shilla graph with b₂ = c₂ and noninteger eigenvalues has the following intersection array:

$$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$

If Γ is a Q-polynomial Shilla graph with b₂ = c₂ and b = 2r, then the graph Γ has intersection array

$$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$

and, for any vertex u in Γ, the subgraph Γ₃(u) is an antipodal distance-regular graph with intersection array

$$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$

The Shilla graphs with b₂ = c₂ and b = 4 are also classified in the paper.

     

A ternary Diophantine inequality by primes with one of the form $$p=x^2+y^2+1$$ p = x 2 + y 2 + 1

The Ramanujan Journal - In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $$1&lt;\frac{427}{400}$$...