Optimal channel assignment and <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, 1)-labeling

The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function \(f:V(G)\rightarrow \{0,1,2,\ldots ,k\}\) such that \(|f(u)-f(v)|\ge p\) if \(d(u,v)=1\) and \(|f(u)-f(v)|\ge 1\) if \(d(u,v)=2\), where d(u, v) is the distance between the two vertices u and v in the graph. Denote \(\lambda _{p,1}^l(G)= \min \{k \mid G\) has a list k-L(p, 1)-labeling\(\}\). In this paper we show upper bounds \(\lambda _{1,1}^l(G)\le \Delta +9\) and \(\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}\) for planar graphs G without 4- and 6-cycles, where \(\Delta \) is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.     

Standard Jordan partitions with three parameters

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let \(J_r\) denote an \(r \times r\) matrix with minimal polynomial \((t-1)^r\) over a field F of characteristic p. For positive integers \(n_1\), \(n_2\), and \(n_3\) satisfying \(n_1 \le n_2 \le n_3\), the Jordan canonical form of the \(n_1 n_2 n_3 \times n_1 n_2 n_3\) matrix \(J_{n_1} \otimes J_{n_2} \otimes J_{n_3}\) has the form \(J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}\) where \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0\) and \(\sum _{i=1}^m \lambda _i=n_1 n_2 n_3\). The partition \(\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)\) of \(n_1 n_2 n_3\), which depends on \(n_1\), \(n_2\), \(n_3\), and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

On complementary dual quasi-twisted codes

A linear complementary-dual (LCD) code C is a linear code whose dual code \(C^{\perp }\) satisfies \(C \cap C^{\perp }=\{0\}\). In this work we characterize some classes of LCD q-ary \((\lambda , l)\)-quasi-twisted (QT) codes of length \(n=ml\) with \((m,q)=1\), \(\lambda \in F_{q} \setminus \{0\}\) and \(\lambda \ne \lambda ^{-1}\). We show that every \((\lambda ,l)\)-QT code C of length \(n=ml\) with \(dim(C)<m\) or \(dim(C^{\perp })<m\) is an LCD code. A sufficient condition for r-generator QT codes is provided under which they are LCD. We show that every maximal 1-generator \((\lambda ,l)\)-QT code of length \(n=ml\) with \(l>2\) is either an LCD code or a self-orthogonal code and a sufficient condition for this family of codes is given under which such a code C is LCD. Also it is shown that every maximal 1-generator \((\lambda ,2)\)-QT code is LCD. Several good and optimal LCD QT codes are presented.     

Random Walks in the Hyperbolic Plane and the Minkowski Question Mark Function

Consider \(G=SL_2(\mathbb {Z})/\{\pm I\}\) acting on the complex upper half plane H by \(h_M(z)=\frac{az\,+\,b}{cz\,+\,d}\) for \(M \in G\). Let \(D=\{z \in H: |z|\ge 1, |\mathfrak {R}(z)|\le 1/2\}\). We consider the set \({\mathcal {E}} \subset G\) with the nine elements M, different from the identity, such that \(\mathrm{tr\,}(MM^T)\le 3\). We equip the tiling of H defined by \(\mathbb {D}=\{h_M(D){:}\, M \in G\}\) with a graph structure where the neighbours are defined by \(h_M(D) \cap h_{M'}(D) \ne \emptyset \), equivalently \(M^{-1}M' \in {\mathcal {E}}\). The present paper studies several Markov chains related to the above structure. We show that the simple random walk on the above graph converges a.s. to a point X of the real line with the same distribution of \(S_2 W^{S_1}\), where \(S_1,S_2,W\) are independent with \(\Pr (S_i=\pm 1)=1/2\) and where W is valued in (0, 1) with distribution \(\Pr (W<w)=\mathbf ? (w)\). Here \(\mathbf ? \) is the Minkowski function. If \(K_1, K_2, \ldots \) are i.i.d with distribution \(\Pr (K_i=n)= 1/2^n\) for \(n=1,2,\ldots \), then \(W= \frac{1}{K_1+\frac{1}{K_2+\ldots }}\): this known result (Isola in Appl Math 5:1067–1090, 2014) is derived again here.     

<Emphasis Type="Italic">b</Emphasis>-Generalized Skew Derivations on Lie Ideals

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, \(F\ne 0\) an b-generalized skew derivation of R, L a non-central Lie ideal of R, \(0\ne a\in R\) and \(n\ge 1\) a fixed integer. In this paper, we prove the following two results:

1. 1.
   If R has characteristic different from 2 and 3 and \(a[F(x),x]^n=0\), for all \(x\in L\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\), the standard identity of degree 4, and there exist \(\lambda \in C\) and \(b\in Q\), such that \(F(x)=bx+xb+\lambda x\), for all \(x\in R\).
    
2. 2.
   If \(\mathrm{{char}}(R)=0\) or \(\mathrm{{char}}(R) > n\) and \(a[F(x),x]^n\in Z(R)\), for all \(x\in R\), then either there exists an element \(\lambda \in C\), such that \(F(x)=\lambda x\), for all \(x\in R\) or R satisfies \(s_4(x_1,\ldots ,x_4)\).
    

     

The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

Let \(G{/}H\) be a compact homogeneous space, and let \(\hat{g}_0\) and \(\hat{g}_1\) be G-invariant Riemannian metrics on \(G/H\). We consider the problem of finding a G-invariant Einstein metric g on the manifold \(G/H\times [0,1]\) subject to the constraint that g restricted to \(G{/}H\times \{0\}\) and \(G/H\times \{1\}\) coincides with \(\hat{g}_0\) and \(\hat{g}_1\), respectively. By assuming that the isotropy representation of \(G/H\) consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.     

Simultaneous power factorization in modules over Banach algebras

Let A be a Banach algebra with a bounded left approximate identity \(\{e_\lambda \}_{\lambda \in \Lambda }\), let \(\pi \) be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that \(\lim _{\lambda }\pi (e_\lambda )s=s\) uniformly on S. If S is bounded, or if \(\{e_\lambda \}_{\lambda \in \Lambda }\) is commutative, then we show that there exist \(a\in A\) and maps \(x_n: S\rightarrow X\) for \(n\ge 1\) such that \(s=\pi (a^n)x_n(s)\) for all \(n\ge 1\) and \(s\in S\). The properties of \(a\in A\) and the maps \(x_n\), as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of \(\mathrm {C}_0({\mathbb R})\) and unbounded S is included.     

On the eigenvalues and spectral radius of starlike trees

Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\), that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\), then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\), where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\).     

Erratum to: Abstract commensurability and quasi-isometry classification of hyperbolic surface group amalgams

A fundamental result by Gromov and Thurston asserts that, if M is a closed hyperbolic n-manifold, then the simplicial volume \(\Vert M\Vert \) of M is equal to \(\mathrm{Vol}(M)/v_n\), where \(v_n\) is a constant depending only on the dimension of M. The same result also holds for complete finite-volume hyperbolic manifolds without boundary, while Jungreis proved that the ratio \(\mathrm{Vol}(M)/\Vert M\Vert \) is strictly smaller than \(v_n\) if M is compact with nonempty geodesic boundary. We prove here a quantitative version of Jungreis’ result for \(n\ge 4\), which bounds from below the ratio \(\Vert M\Vert /\mathrm{Vol}(M)\) in terms of the ratio \(\mathrm{Vol}(\partial M)/\mathrm{Vol}(M)\). As a consequence, we show that, for \(n\ge 4\), a sequence \(\{M_i\}\) of compact hyperbolic n-manifolds with geodesic boundary satisfies \(\lim _i \mathrm{Vol}(M_i)/\Vert M_i\Vert =v_n\) if and only if \(\lim _i \mathrm{Vol}(\partial M_i)/\mathrm{Vol}(M_i)=0\). We also provide estimates of the simplicial volume of hyperbolic manifolds with geodesic boundary in dimension 3.     

Revisiting Fryszkowski’s problem

Professor Andrzej Fryszkowski formulated, at the 2nd Symposium on Nonlinear Analysis in Toruń, September 13–17, 1999, the following problem: given \(\alpha \in (0,1)\), an arbitrary non-empty set \(\Omega \) and a set-valued mapping \(F:\Omega \rightarrow 2^{\Omega }\), find necessary and (or) sufficient conditions for the existence of a (complete) metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d. Com?neci (Stud. Univ. Babe?-Bolyai Math. 62:537–542, 2017) provided necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) having the property that F is a Nadler set-valued \(\alpha \)-contraction with respect to d, in case that \(\alpha \in (0,\frac{1}{2})\) and there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) . We improve Com?neci’s result by allowing \(\alpha \) to belong to the interval (0, 1). In addition, we provide necessary and sufficient conditions for the existence of a complete and bounded metric d on \(\Omega \) such that F is a Nadler set-valued \(\alpha \)-similarity with respect to d, in case that \(\alpha \in (0,1)\), there exists \(z\in \Omega \) such that \(F(z)=\{z\}\) and F is non-overlapping.     

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics

$$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$

which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

     

Forcing Posets with Large Dimension to Contain Large Standard Examples

The dimension of a poset P, denoted \(\dim (P)\), is the least positive integer d for which P is the intersection of d linear extensions of P. The maximum dimension of a poset P with \(|P|\le 2n+1\) is n, provided \(n\ge 2\), and this inequality is tight when P contains the standard example \(S_n\). However, there are posets with large dimension that do not contain the standard example \(S_2\). Moreover, for each fixed \(d\ge 2\), if P is a poset with \(|P|\le 2n+1\) and P does not contain the standard example \(S_d\), then \(\dim (P)=o(n)\). Also, for large n, there is a poset P with \(|P|=2n\) and \(\dim (P)\ge (1-o(1))n\) such that the largest d so that P contains the standard example \(S_d\) is o(n). In this paper, we will show that for every integer \(c\ge 1\), there is an integer \(f(c)=O(c^2)\) so that for large enough n, if P is a poset with \(|P|\le 2n+1\) and \(\dim (P)\ge n-c\), then P contains a standard example \(S_d\) with \(d\ge n-f(c)\). From below, we show that \(f(c)={\varOmega }(c^{4/3})\). On the other hand, we also prove an analogous result for fractional dimension, and in this setting f(c) is linear in c. Here the result is best possible up to the value of the multiplicative constant.     

A generalization of the normal rational curve in $$\mathop {\mathrm{PG}}(d,q^n)$$ and its associated non-linear MRD codes

Let A and B be two points of \(\mathop {\mathrm{PG}}(d,q^n)\) and let \(\Phi \) be a collineation between the stars of lines with vertices A and B, that does not map the line AB into itself. In this paper we prove that if \(d=2\) or \(d\ge 3\) and the lines \(\Phi ^{-1}(AB), AB, \Phi (AB) \) are not in a common plane, then the set \(\mathcal{C}\) of points of intersection of corresponding lines under \(\Phi \) is the union of \(q-1\) scattered \({\mathbb {F}}_{q}\)-linear sets of rank n together with \(\{A,B\}\). As an application we will construct, starting from the set \(\mathcal{C}\), infinite families of non-linear \((d+1, n, q;d-1)\)-MRD codes, \(d\le n-1\), generalizing those recently constructed in Cossidente et al. (Des Codes Cryptogr 79:597–609, 2016) and Durante and Siciliano (Electron J Comb, 2017).     

Existence of positive solution for nonlocal singular fourth order Kirchhoff equation with Hardy potential

This paper is concerned with the existence of positive solution to a class of singular fourth order elliptic equation of Kirchhoff type

$$\begin{aligned} \triangle ^2 u-\lambda M(\Vert \nabla u\Vert ^2)\triangle u-\frac{\mu }{\vert x\vert ^4}u=\frac{h(x)}{u^\gamma }+k(x)u^\alpha , \end{aligned}$$

under Navier boundary conditions, \(u=\triangle u=0\). Here \(\varOmega \subset {\mathbf {R}}^N\), \(N\ge 1\) is a bounded \(C^4\)-domain, \(0\in \varOmega \), h(x) and k(x) are positive continuous functions, \(\gamma \in (0,1)\), \(\alpha \in (0,1)\) and \(M:{\mathbf {R}}^+\rightarrow {\mathbf {R}}^+\) is a continuous function. By using Galerkin method and sharp angle lemma, we will show that this problem has a positive solution for \(\lambda > \frac{\mu }{\mu ^*m_0}\) and \(0<\mu <\mu ^*\). Here \(\mu ^*=\Big (\frac{N(N-4)}{4}\Big )^2\) is the best constant in the Hardy inequality. Besides, if \(\mu =0\), \(\lambda >0\) and h, k are Lipschitz functions, we show that this problem has a positive smooth solution. If \(h,k\in C^{2,\,\theta _0}(\overline{\varOmega })\) for some \(\theta _0\in (0,1)\), then this problem has a positive classical solution.

     

A note on amicable numbers and their variations

For any positive integer n, let \(\sigma (\mathrm{n})\) and p(n) denote the sum of divisors and the least prime divisor of n respectively. Let a, b be positive integers. In this paper we prove the following two results: (i) If 4 | a and \(\gcd (a, b)=1\), then a and b do not satisfy \(\sigma (a)= \sigma (b)=a+b\). (ii) If \(a>10^{8}\) and \(p(a)>2\log _{2}a+1\), where \(\log _{2}{a}\) is the logarithm of a with base 2, then a and b do not satisfy \(\sigma (a)=\sigma (b)=a+b+\lambda \), where \(\lambda \in \{0,\pm 1\}\).     

Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

Given a connected simple graph \(G=(V(G),E(G))\), a set \(S\subseteq V(G)\) is said to be a 2-metric generator for G if and only if for any pair of different vertices \(u,v\in V(G)\), there exist at least two vertices \(w_1,w_2\in S\) such that \(d_G(u,w_i)\ne d_G(v,w_i)\), for every \(i\in \{1,2\}\), where \(d_G(x,y)\) is the length of a shortest path between x and y. The minimum cardinality of a 2-metric generator is the 2-metric dimension of G, denoted by \(\dim _2(G)\). The metric \(d_{G,2}: V(G)\times V(G)\longmapsto {\mathbb {N}}\cup \{0\}\) is defined as \(d_{G,2}(x,y)=\min \{d_G(x,y),2\}\). Now, a set \(S\subseteq V(G)\) is a 2-adjacency generator for G, if for every two vertices \(x,y\in V(G)\) there exist at least two vertices \(w_1,w_2\in S\), such that \(d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)\) for every \(i\in \{1,2\}\). The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by \({\mathrm {adim}}_2(G)\). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product \(G\circ H\) of two graphs G and H. Specifically, we show that \(\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),\) where \(f(G,H)\ge 0\), and determine all the possible values of f(G, H).     

Classical and Free Fourth Moment Theorems: Universality and Thresholds

Let \(X\) be a centered random variable with unit variance and zero third moment, and such that \(\mathrm{IE}[X^4] \ge 3\). Let \(\{F_n {:}\, n\ge 1\}\) denote a normalized sequence of homogeneous sums of fixed degree \(d\ge 2\), built from independent copies of \(X\). Under these minimal conditions, we prove that \(F_n\) converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to \(3\). The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit thresholds, as well as establish several connections with the so-called universality phenomenon of probability theory. Both in the classical and free probability frameworks, our results extend and unify previous Fourth Moment Theorems for Gaussian and semicircular approximations. Our techniques are based on a fine combinatorial analysis of higher moments for homogeneous sums.     

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.