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In this paper, we construct new self-dual codes from a construction that involves a unique combination; \begin{document}$ 2 \times 2 $\end{document} block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields self-dual codes. The theory is supported by the construction of self-dual codes over the rings \begin{document}$ \mathbb{F}_2 $\end{document} , \begin{document}$ \mathbb{F}_2+u \mathbb{F}_2 $\end{document} and \begin{document}$ \mathbb{F}_4+u \mathbb{F}_4 $\end{document} . Using extensions and neighbours of codes, we construct \begin{document}$ 32 $\end{document} new self-dual codes of length \begin{document}$ 68 $\end{document} . We construct 48 new best known singly-even self-dual codes of length 96.

Given a graph G and a set S ⊆ V ( G ) , the Δ -interval of S , [ S ] Δ , is the set formed by the vertices of S and every w ∈ V ( G ) forming a triangle with two vertices of S . If [ S ] Δ = S , then S is Δ -convex of G ; if [ S ] Δ = V ( G ) , then S is a Δ -interval set of G . The Δ -interval number of G is the minimum cardinality of a Δ -interval set and the Δ -convexity number of G is the maximum cardinality of a proper Δ -convex subset of V ( G ) . In this work, we show that the problem of computing the Δ -convexity number is W[1]-hard and NP-hard to approximate within a factor O ( n 1 − ɛ ) for any constant ɛ > 0 even for graphs with diameter 2 and that the problem of computing the Δ -interval number is NP-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the Δ -convexity number of chordal graphs and for computing the Δ -interval number of block graphs. We also present results on the Δ -hull, Δ -interval and Δ -convexity numbers concerning the three standard graph products, namely, the Cartesian, the strong and the lexicographic products, in function of these and well-studied parameters of the operands.

An Italian dominating function on a simple undirected graph G is a function f : V ( G ) ⟶ { 0 , 1 , 2 } satisfying the condition that for each vertex v with f ( v ) = 0 , ∑ u ∈ N G ( v ) f ( u ) ≥ 2 . An Italian dominating function f on G is called a perfect Italian dominating function on G if for each vertex v with f ( v ) = 0 , ∑ u ∈ N G ( v ) f ( u ) = 2 . The weight of a function f on a graph G , denoted by w ( f ) , is the sum ∑ v ∈ V ( G ) f ( v ) . For a simple undirected graph G , Min-PIDF is the problem of finding the minimum weight of a perfect Italian dominating function on G . First, we discuss the complexity difference between Min-PIDF and the problem of finding the minimum weight of an Italian dominating function. We then establish the NP -completeness of the decision version of Min-PIDF in chordal graphs and investigate the hardness of approximation of Min-PIDF in general graphs. Finally, we present linear time algorithms for computing the minimum weight of a perfect Italian dominating function in block graphs and series-parallel graphs.

The nullity of a vertex-weighted graph is equal to the number of zero eigenvalues of its adjacency matrix. In this article, we give a linear time algorithm to calculate the nullity of vertex-weighted block graphs, i.e., graphs in which every block is a clique, with weights assigned to the vertices.

We extend the block filtration, defined by Brown based on the work of Charlton, to all motivic multiple zeta values, and study relations compatible with this filtration. We construct a Lie algebra describing relations among motivic multiple zeta values modulo terms of lower block degree, proving Charlton's cyclic insertion conjecture in this structure, and showing the existence of a `block shuffle' relation, and a previously unknown dihedral symmetry and differential relation., Comment: 23 pages, based on work in the author's doctoral thesis. Typo in statement of Corollary 2.9 corrected

Let $\mathbf{H}$ be the cartesian product of a family of finite abelian groups indexed by a finite set $\Omega$. A given poset (i.e., partially ordered set) $\mathbf{P}=(\Omega,\preccurlyeq_{\mathbf{P}})$ gives rise to a poset metric on $\mathbf{H}$, which further leads to a partition $\mathcal{Q}(\mathbf{H},\mathbf{P})$ of $\mathbf{H}$. We prove that if $\mathcal{Q}(\mathbf{H},\mathbf{P})$ is Fourier-reflexive, then its dual partition $\Lambda$ coincides with the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$, the dual poset of $\mathbf{P}$, and moreover, $\mathbf{P}$ is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in \cite{4}. We also show that with some other assumptions, $\Lambda$ is finer than the partition of $\hat{\mathbf{H}}$ induced by $\mathbf{\overline{P}}$. In addition, we give some necessary and sufficient conditions for $\mathbf{P}$ to be hierarchical, and for the case that $\mathbf{P}$ is hierarchical, we give an explicit criterion for determining whether two codewords in $\hat{\mathbf{H}}$ belong to the same block of $\Lambda$. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.

Let B ( n , q ) be the class of block graphs on n vertices having all their blocks of the same size. We prove that if G ∈ B ( n , q ) has at most three pairwise adjacent cut vertices then the minimum spectral radius ρ ( G ) is attained at a unique graph. In addition, we present a lower bound for ρ ( G ) when G ∈ B ( n , q ) .

Background: Interfascial plane block can be used to treat postoperative pain after laparoscopic surgery. This study aimed to investigate the effect of ultrasound-guided unilateral rhomboid intercostal and subserratus plane (RISS) block after laparoscopic cholecystectomy on the amount of analgesic consumption.Methods: Fifty patients who underwent laparoscopic cholecystectomy were included in this quasi-experimental study. Patients fulfilling the inclusion criteria were analyzed in two groups: RISS group (RISS block with 20 ml of 0.25% bupivacaine + intravenous patient-controlled analgesia [IV-PCA] tramadol [n = 25]); and Control group (IV-PCA tramadol [n = 25]). The primary outcome was the total amount of tramadol used over 24 h. Secondary outcomes included side effects, additional analgesic use, and postoperative pain (at rest and during activity) at 2, 6, 12, and 24 h according to numerical rating scale (NRS) scores.Results: Postoperative tramadol consumption at 24 h was significantly lower in the RISS group than in the Control group (P < 0.001). Resting NRS scores at 2 h and 6 h were significantly lower in the RISS group. NRS scores during movement in the RISS group were significantly lower at 2, 6, and 12 h postoperatively. There was no statistically significant difference in the rate of side effects and additional analgesic use between the groups (P > 0.05). Conclusions: Unilateral RISS block was an effective method for pain management after laparoscopic cholecystectomy and can be used as a part of multimodal analgesia.

The aim of this study was to compare the closed reduction interfragmentary pinning method (IPM) with the extension block technique (EBT) for bony mallet finger. Patients who underwent mallet finger operations were screened retrospectively for the following inclusion criteria: Doyle type 4c, age between 18 and 75 years, less than 4 weeks to surgery, and more than 1 year of follow-up time. Group I underwent a closed reduction IPM, and group II underwent the EBT. Lateral radiographs taken during the preoperative and final examination were used to evaluate the size and amount of displacement from the distal interphalangeal (DIP) joint and the dorsal fragment as well as the articular surface. Operation times were compiled from patient records. During the final examination, pain and DIP joint range of motion (ROM) were assessed and complications were recorded. The Crawford criteria were used for functional results. Fifteen patients in group I (8 men, 7 women) and 17 patients in group II (10 men, 7 women) were evaluated. Age, gender, time to surgery and follow-up time showed no statistically significant differences between the two groups. The differences in fragment size, preoperative and postoperative joint displacement, amount of dorsal displacement and DIP joint ROM were not statistically significant between the two groups. However, the operation time was significantly shorter time in group I than in group II (p=0.000). The average time to fracture union was significantly longer in group I (7.3 weeks) than in group II (6 weeks) (p=0.013). The EBT has faster time to union and is a safer method with lesser risk of arthritis and fragmentation. The IPM can be an alternative with shorter operation time, less pin bed infection and nail bed damage, especially in Doyle type 4c cases with large fragments.

Objective To investigate the effects of Erector Spinae Plane Block (ESPB) and Retrolaminar Block (RLB) on intra- and postoperative analgesia in patients with Multiple Rib Fractures (MRFs). Methods A total of 80 MRFs patients were randomly divided into the ESPB (Group E) and RLB (Group R) groups. After general anesthesia, ESPB and RLB were performed under ultrasound guidance, respectively, together with 20 mL of 0.5% ropivacaine and Patient-Controlled Intravenous Analgesia (PCIA). Results Thirty-four cases in Group E and 33,cases in Group R showed unclear paravertebral spaces. The intraoperative dosage of remifentanil (mean ± SD) (392.8 ± 118.7 vs. 501.7 ± 190.0 μg) and postoperative morphine PCIA dosage, (7.35 ± 1.55 vs. 14.73 ± 2.18 mg) in Group R were significantly less than those in Group E; the Visual Analog Scale (VAS) scores in Group R at 2 (2.7 ± 1.2 vs. 3.4 ± 1.4), 4 (2.2 ± 1.1 vs. 2.8 ± 0.9), 12 (2.5 ± 0.9 vs. 3.0 ± 0.8), and 24 hours (2.6 ± 1.0 vs. 3.1 ± 0.9) after surgery were significantly lower than those in Group E. Finally, the normal respiratory diaphragm activity (2.17 ± 0.22 vs. 2.05 ± 0.19), pH (median [IQR] (7.38 [7.31–7.45] vs. 7.36 [7.30–7.42]), and partial pressure of carbon dioxide (PaCO2) (44 [35–49] vs. 42.5 [30–46]) after the operation in Group R were significantly better than those in Group E (p Conclusions RLB was a more effective analgesic method than ESPB in the treatment of MRF.

Let G = S L ( 2 , 5 ) be the special linear group of 2 × 2 -matrices with coefficients in the field with 5 elements. We show that the principal block over a splitting field K of characteristic two of the group algebra KG has a 3-cluster tilting module. This gives the first example of a representation-infinite block of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm.

Feedback shift registers (FSRs), which have two configurations: Fibonacci and Galois, are a primitive building block in stream ciphers. In this paper, a transformation between Fibonacci FSRs and Galois FSRs is improved based on semi-tensor product (STP) of matrices. It is verified that a weakly equivalent Galois FSR with fewer stages cannot be found for a Fibonacci FSR with n stages, but not vice versa. Furthermore, for a given Fibonacci FSR with n stages, there are totally ( 2 n − 1 ) ! 2 − 1 weakly equivalent Galois FSRs. Additionally, an effective algorithm is developed to reduce the number of variables of the Galois FSRs while keeping it weakly equivalent to the given Fibonacci FSR. Finally, the feasibility of the proposed strategies is demonstrated by numerical examples.

The digital nerve block is an effective method of anesthesia before minor surgical interventions on the fingers. However, patients may experience a lot of pain and anxiety during this procedure. The efficacy of topical vapocoolant spray during minor procedures has been demonstrated in previous studies. we aimed to evaluate the effectiveness of topical vapocoolant spray in reducing pain during digital nerve block.This prospective, randomized clinical study was conducted to evaluate the effectiveness of vapocoolant spray application in reducing pain during digital block. The patients were categorized into 2 groups as spray-treated and control group. A routine digital block process was applied to the control group. Spray application was performed in two groups of 50 each, in a manner of bilateral and unilateral to the finger. Demographic data of the patients, such as gender, age, dominant hand, injury patterns, injection site and injury sites, were recorded. After the application, the patients' visual analog scale (VAS) was evaluated.Of the participants, 100 were randomly assigned to the vapocoolant spray-treated group, and 50 were included in the control group. The VAS pain score during penetration in both spray groups was significantly lower than the control group (p0.001). Pain change during penetration was found to be significantly lower in the bilateral spray-treated group compared to the control group (p0.001). Pain change during infiltration was significantly lower in both spray groups compared to the control group (p0.001).Spray application prior to digital nerve blocking can be used to reduce needle penetration pain associated with this procedure and pain associated with local anesthetic infiltration.

The aim of this paper is to describe the structure of the finitely generated subgroups of a family of branch groups, which includes the first Grigorchuk group and the Gupta-Sidki 3-group. This description is made via the notion of block subgroup. We then use this to show that all groups in the above family are subgroup separable (LERF). These results are obtained as a corollary of a more general structural statement on subdirect products of just infinite groups., 22 pages, 2 figures; V3: Final version, to appear in Izvestiya: Mathematics

We consider the problem of communicating over a channel that randomly “tears” the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\mathrm{ Geometric}}(p_{n})$ , we characterize the capacity as $C = e^{-\alpha }$ , where $\alpha = \lim _{n\to \infty } p_{n} \log n$ . Our results show that the case of ${\mathrm{ Geometric}}(p_{n})$ -length fragments and the case of deterministic length- $(1/p_{n})$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity.

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint t-cliques. The extremal graphs attaining the bounds are exactly the block graphs of Steiner 2-designs and the regular graphs with K t -decompositions, respectively.

Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix $$\rho (P_{GS})\ge \rho (P_S)\ge \rho (P_{AGS})$$ , where $$P_{GS}, P_S, P_{AGS}$$ are the iteration matrices of the Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.

Major lumbar spine surgery causes severe pain in the postoperative period. There are few studies regarding the effect of erector spinae plane block (ESPB) effect on lumbar surgery and its effect is still controversial. Therefore, the study aimed to investigate the effect of ultrasound-guided low thoracic ESPB on opioid consumption and postoperative pain score. Seventy-eight patients undergoing elective open lumbar spine surgery were randomized into two groups. In ESPB group (n = 35) received ultrasound-guided ESPB and in the control group (n = 35), there was no block. Postoperative opioid consumption as morphine equivalent dose, numerical rating scale, mobilization time, discharge time and side effects, bolus deliveries, rescue analgesia doses were evaluated. Total opioid consumption as morphine equivalent was higher in the control group than the ESPB group (p = 0.000). Compare with the control group, the numeric rating scale scores were lower in the ESPB group at the 6th, 12th, and 24th hours (p

In this paper, we study enumeration formulas of $${\mathbb {Z}}_4$$ -double cyclic codes. We give the enumerations of separable $${\mathbb {Z}}_4$$ -double cyclic codes and a class of non-separable $${\mathbb {Z}}_4$$ -double cyclic codes with the coprime block lengths.

This work was aimed to study the analgesic effect of pudendal nerve block on obstetrics and gynecology under the guidance of ultrasound image based on optimized fast super resolution reconstructed convolutional neural network (FSRCNN) algorithm. A total of 110 primiparas from hospital who gave birth through vagina were randomly rolled into experimental group (55 cases) and control group (55 cases). The optimized FSRCNN algorithm was constructed, compared with the FSRCNN algorithm and the Bicubic algorithm and applied to 110 cases of maternal patients undergoing perineotomy under ultrasound image-guided pudendal nerve block. Visual analogue scoring (VAS), incision suture pain VAS score, occurrence of complications, puerpera labor time, and newborn weight were recorded and compared, so did Apgar score of newborns, numbness of maternal thigh, recovery of puncture site, and satisfaction of maternal analgesia. The results showed that the peak signal-to-noise ratio (PSNR) of the high-resolution image reconstructed by the FSRCNN algorithm was 32.68 dB and that reconstructed by the optimized FSRCNN algorithm was 32.19 dB. The PSNR of the Bicubic algorithm processed image was 28.51 dB. In the lateral resection of episiotomy in the second stage of labor, the visual analog score (2.3 ± 1.5 points) of the experimental group was inferior to that of the control group (7.1 ± 2.6 points) ( P < 0.05 ). The visual analogue score of stitch pain (1.3 ± 0.8 points) was also inferior to that of the control group (5.2 ± 1.9 points) ( P < 0.05 ). Moreover, the satisfaction of the parturients in the experimental group (9.86 ± 0.41 points) was considerably superior to that of the control group (7.36 ± 1.25 points) ( P < 0.05 ). In short, the optimized FSRCNN algorithm had a short training time and good reconstruction effect. Ultrasound-guided pudendal nerve block had a substantial analgesic effect on the second stage of labor and improved parturients’ satisfaction.

The main goal of this paper is to elaborate some results on the spectral properties of $$2 \times 2$$ block matrix linear relations with unbounded entries and with a domain consisting of vectors which satisfy certain relations between their components. We present some conditions to prove some Frobenius–Schur decompositions for linear relations and characterize the stability of the essential spectra of these linear relations. Our results generalize many known ones in the literature, in particular those obtained by Alvarez et al. (in Math Methods Appl Sci 37:620–644, 2014) and Batkai et al. (in Math Nachr 278:1408–1429, 2005).

A radio labeling of a graph $G$ is a mapping $f$ : $V(G) \rightarrow \{0, 1, 2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ holds for every pair of vertices $u$ and $v$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest $t$ such that $G$ admits a radio labeling with $t=\max\{|f(v)-f(u)|: v, u \in V(G)\}$. A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is established. The block graph which achieves this bound is called a lower bound block graph. We prove three necessary and sufficient conditions for lower bound block graphs. Moreover, we give three sufficient conditions for a graph to be a lower bound block graph. Applying the established bound and conditions, we show that several families of block graphs are lower bound block graphs, including the level-wise regular block graphs and the extended star of blocks. The line graph of a graph $G(V,E)$ has $E(G)$ as the vertex set, where two vertices are adjacent if they are incident edges in $G$. We extend our results to trees as trees and its line graphs are block graphs. We prove that if a tree is a lower bound block graph then, under certain conditions, its line graph is also a lower bound block graph, and vice versa. Consequently, we show that the line graphs of many known lower bound trees, excluding paths, are lower bound block graphs., 21 pages, 4 figures. This is the final version accepted in Theoretical Computer Science(TCS) Journal

Introduction The aim of this study is to verify if the shoulder anterior capsular block (SHAC), combined with other nerve blocks, is effective in relieving shoulder pain, avoiding motor block and allowing an early rehabilitation program. Methods Seventy-five consecutive patients with painful shoulder were treated with the SHAC, alone (30 patients) or in combination with a suprascapular nerve block (SSnb: 25 patients) or with pectoralis and serratus plane block (PECS-2: 20 patients). All blocks were performed with 0.2% ropivacaine plus 8 mg dexamethasone. All patients were treated with three-weekly physiotherapy sessions for the following 2 weeks and then with home exercises. Results The post-procedural analgesic effect was strong in all groups, with a mean change in numeric rating scale (NRS) values of −6.05 in group 1, −6.25 in group 2, and −6.19 in group 3 (p

Introduction Postoperative analgesia in elderly patients is still a thorny problem. Ultrasound-guided oblique subcostal transverse abdominis plane block (TAPB) has been demonstrated to provide postoperative analgesia after abdominal surgeries. However, recent studies have suggested that an alternative method, erector spinae plane block (ESPB), might also be effective. In this study, we compared the postoperative analgesic effects of ESPB and TAPB in elderly patients who had undergone laparoscopic colorectal surgery. Methods Sixty-two elderly patients (≥ 65 years old) scheduled for elective laparoscopic colorectal surgery with general anesthesia were randomly allocated to two equally sized groups: ESPB group and TAPB group. The ESPB group had a bilateral erector spinae plane block, and the TAPB group had a bilateral oblique subcostal transverse abdominis plane block. The primary outcome was visual analogue scale (VAS) pain score during the first 24 postoperative hours at resting and active states. The secondary outcomes were postoperative consumption of sufentanil, satisfaction score, the number of patients who required antiemetics, incidence of block-related complications, and other side events. Results There were no demographic differences between two groups. Compared to the TAPB group, the ESPB group had lower VAS pain scores and sufentanil consumption during the first 24 postoperative hours. Additionally, ESPB reduced the occurrence of postoperative nausea and vomiting. Furthermore, the satisfaction score was higher in the ESPB group. No other complications were reported between the two groups. Conclusions Compared with oblique subcostal TAPB, ESPB more effectively reduced postoperative pain and opioid consumption. Thus, ESPB is suitable for postoperative analgesia in elderly patients who have undergone laparoscopic colorectal surgery. Trial Registration Chinese Clinical Trial Registry: ChiCTR2000033236. Supplementary Information The online version contains supplementary material available at 10.1007/s40122-021-00329-x.

In this article, we investigate $2$-$(v,k,\lambda)$ designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$., Comment: arXiv admin note: text overlap with arXiv:1904.10518

The number of Clar covers, the number of Kekule structures, and the Clar covering polynomials (aka Zhang–Zhang or ZZ polynomials) of benzenoid parallelogram chains M k m , n formed by merging k benzenoid parallelograms M m , n are characterized in terms of analogous quantities of the elementary building block, M m , n . The appropriate formulas are compactly expressed as determinants of highly structured, tridiagonal, Toeplitz k × k matrices. All the 2 k distinct parallelogram chains M k m , n ≡ M 1 M 2 … M k of constant length k , where M i ∈ R ≡ M m , n , L ≡ M n , m , share the same ZZ polynomial and consequently possess the same number of Clar covers and Kekule structures. The presented results constitute the first attempt to express the Clar theory of complex benzenoid moieties in terms of elementary benzenoids.

In this work, we present block diagonalization and power allocation algorithms for large-scale multiple-antenna systems with coarsely quantized signals. In particular, we develop Coarse Quantization-Aware Block Diagonalization ${\scriptstyle \mathrm {\left ({CQA-BD}\right)}}$ and Coarse Quantization-Aware Regularized Block Diagonalization ${\scriptstyle \mathrm {\left ({CQA-RBD}\right)}}$ precoding algorithms that employ the Bussgang decomposition and can mitigate the effects of low-resolution signals and interference. Moreover, we also devise the Coarse Quantization-Aware Most Advantageous Allocation Strategy ${\scriptstyle \mathrm {\left ({CQA-MAAS}\right)}}$ power allocation algorithm to improve the sum rate of precoders that operate with low-resolution signals. An analysis of the sum-rate performance is carried out along with computational complexity and power consumption studies of the proposed and existing techniques. Simulation results illustrate the performance of the proposed ${\scriptstyle \mathrm {CQA-BD}}$ and ${\scriptstyle \mathrm {CQA-RBD}}$ precoding algorithms, and the proposed ${\scriptstyle \mathrm {CQA-MAAS}}$ power allocation strategy against existing approaches.

Finite metric spaces are an essential tool in discrete mathematics and have applications in several areas including computational biology, image analysis, speech recognition, and information retrieval. Given any such metric D on a finite set X , an important problem is to find appropriate ways to realize D by weighting the edges in some graph G containing X in its vertex set such that D ( x , y ) equals the length of a shortest path from x to y in G for all x , y ∈ X . Here we focus on realizations with minimum total edge weight, called optimal realizations. By considering the 2-connected components and bridges in any optimal realization G of D we obtain an additive decomposition of D into simpler metrics. We show that this decomposition, called the block decomposition, is canonical in that it only depends on D and not on G , and that the decomposition can be computed in O ( | X | 3 ) time. As well as providing a fundamental new way to decompose any finite metric space, we expect that the block decomposition will provide a useful preprocessing tool for deriving metric realizations.

The attackers in a network may have a tendency of targeting on a group of clustered nodes, and they hope to avoid the existence of significant large communication groups in the remaining network, such as botnet attack, DDoS attack, and Local Area Network Denial attack. Current various kinds of connectivity do not well reflect the fault tolerance of a network under these attacks. This observation inspires a new measure for network reliability to resist the block attack by taking into account of the dispersity of the remaining nodes. Let $G$ G be a network, $C\subset V(G)$ C ⊂ V ( G ) and $G[C]$ G [ C ] be a connected subgraph . Then $C$ C is called an $h$ h -faulty-block of $G$ G if $G-C$ G - C is disconnected, and every component of $G-C$ G - C has at least $h+1$ h + 1 nodes. The minimum cardinality over all $h$ h -faulty-blocks of $G$ G is called $h$ h -faulty-block connectivity of $G$ G , denoted by ${FB}\kappa _h(G)$ F B κ h ( G ) . In this article, we determine ${FB}\kappa _h(Q_n)$ F B κ h ( Q n ) for $n$ n -dimensional hypercube $Q_n$ Q n ( $n\geq 4$ n ≥ 4 ), a classic interconnection network. We establish that ${FB}\kappa _h(Q_n)=(h+2)n-3h-1$ F B κ h ( Q n ) = ( h + 2 ) n - 3 h - 1 for $0\leq h\leq 1$ 0 ≤ h ≤ 1 , and ${FB}\kappa _h(Q_n)=(h+2)n-4h+1$ F B κ h ( Q n ) = ( h + 2 ) n - 4 h + 1 for $2\leq h\leq n-2$ 2 ≤ h ≤ n - 2 , respectively. Larger $h$ h -faulty-block connectivity implies that an attacker will have to stage an attack to a bigger block of connected nodes, so that each remaining components will not be too small, which will in turn limit the size of large components. In other words, there will not be great disparity in sizes between any two remaining components, and hence there will less likely be a significantly large remaining communication group. The larger the $h$ h -faulty-block, the more difficult for an attacker to achieve that goal. As a consequence, the resistance of the network against the attacker will increase. Our experiments also show that as $h$ h increases, the $h$ h -faulty-block gets larger, and the size disparity between any two remaining components decreases. In turn, as expected, the size of the largest remaining communication group becomes smaller.

Several constructions of Mutually Unbiased Bases (MUBs) borrow tools from combinatorial objects. In this paper we focus on how one can construct Approximate Real MUBs (ARMUBs) with improved parameters using results from the domain of Resolvable Block Designs (RBDs). We first explain the generic idea of our strategy in relating the RBDs with MUBs/ARMUBs, which are sparse (the basis vectors have small number of non-zero co-ordinates). Then specific parameters are presented, for which we can obtain new classes and improve the existing results. To be specific, we present an infinite family of $\lceil \sqrt {d}\rceil $ many ARMUBs for dimension d = q(q + 1), where q ≡ 3 mod 4 and it is a prime power, such that for any two vectors v1,v2 belonging to different bases, $|\langle {v_{1}|v_{2}}\rangle | < \frac {2}{\sqrt {d}}$ . We also demonstrate certain cases, such as d = sq2, where q is a prime power and sq ≡ 0 mod 4. These findings subsume and improve our earlier results in [Cryptogr. Commun. 13, 321-329, January 2021]. This present construction idea provides several infinite families of such objects, not known in the literature, which can find efficient applications in quantum information processing for the sparsity, apart from suggesting that parallel classes of RBDs are intimately linked with MUBs/ARMUBs.

Number pyramids are common in elementary school mathematics. Trying to express the value of the top block in terms of the values at the base leads to the binomial coefficients. It also seems natural to ask for the maximal number of odd numbers in a number pyramid of a given size. The answer is easy to state, but the proof is nontrivial: A $$k$$ k step number pyramid can have at most $$\left\lfloor\frac{k(k+1)+1}{3}\right\rfloor$$ k ( k + 1 ) + 1 3 odd numbers, which equals two thirds of the number of blocks rounded to the nearest integer. All maximal and almost maximal solutions are given explicitly. To this end, we rephrase the question in terms of colored tilings. In the outlook we present relations to other—mostly geometric—subjects and problems.

For an ordering of the blocks of a design, the point sum of an element is the sum of the indices of blocks containing that element. Block labelling for popularity asks for the point sums to be as equal as possible. For Steiner systems of order v and strength t in general, the average point sum is $$O(v^{2t-1})$$ ; under various restrictions on block partitions of the Steiner system, the difference between the largest and smallest point sums is shown to be $$O(v^{(t+1)/2}\log v)$$ . Indeed for Steiner triple systems, direct and recursive constructions are given to establish that systems exist with all point sums equal for more than two thirds of the admissible orders.

Background: Prostate biopsy is a painful procedure, and the degree of pain is related to the number of core biopsies taken. Objective: To compare the analgesic properties of hyperbaric bupivacaine 0.25% with 0.375% ropivacaine for saddle block in transrectal prostate biopsy. Methods: This was a randomised double-blinded study. Eighty patients with indications for prostate biopsy presenting at the Day-Case Theatre in a Nigerian tertiary facility were randomised into two equal groups: B (Bupivacaine) and R (Ropivacaine). Group B received 1ml of 0.25% bupivacaine, while Group R received 1ml of 0.375% ropivacaine for saddle block, respectively. Pain assessment, home readiness, patients' satisfaction, and time to first analgesic request were assessed and compared between the two groups. Results: The Bupivacaine group had an earlier onset of sensory block (11.90±4.10 minutes vs 23.70±8.65 minutes, p = 0.000), slower sensory block regression (48.73±9.32 minutes vs 24.88±4.21 minutes, p = 0.000), but delayed home readiness (47.23±15.93 minutes vs 29.88±8.58 minutes, p = 0.000), than patients in the Ropivacaine group. The pain scores during, immediately after and 30 minutes post-biopsy were lower in the Bupivacaine group: p = 0.010, p = 0.028 and p = 0.023 respectively. The time to first analgesic request was also longer in the Bupivacaine group (48.73±9.33 minutes) than for those in the Ropivacaine group (24.88±4.21 minutes) with statistical significance (p = 0.000). Conclusion: Intraoperative analgesic properties were better in the Bupivacaine group than in the Ropivacaine group. However, home readiness was earlier in the Ropivacaine group.

Background The purpose of this study was the assessment of the analgesic and hemodynamic implications of dexmedetomidine used as an additive to bupivacaine in ultrasound-guided serratus anterior plane (SAP) block for patients undergoing video-assisted thoracoscopic surgeries (VATS ) under general anesthesia. Results The hemodynamic stability was maintained perioperatively with no significant difference of MBP and HR recordings between the two study groups (P > 0.05). The time to 1st postoperative analgesic demand was significantly longer in group BD than in group B (P < 0.05). The postoperative total nalbuphine and rescue ketorolac requirements were significantly lower in group BD than in group B (P P > 0.05). Ramsay sedation scores were significantly higher in the group BD than in group B in the initial 1st h after surgery (P < 0.05) with no significant difference at the subsequent postoperative recordings (P > 0.05). Conclusion Using dexmedetomidine (0.5 μg/kg) as an additive to bupivacaine for SAP block prolongs the duration of postoperative analgesia and reduces the postoperative analgesic requirements in the 1st 24 h after VATS without any significant side effects.

Background: Abdominoplasty is a popular aesthetic procedure. Both patients and doctors are concerned about post-operative pain and discomfort. erector spinae plane block (ESP) with ultrasound guidance (UG) was an excellent analgesic technique in bariatric surgery. Objectives; to evaluate the intensity and duration of analgesia (ESP) technique to that provided by the transversus abdominis plane (TAP) technique were done bilaterally with (UG) post abdominoplasty. Study Design; A single-blinded, clinical, prospective, randomized study. Patients and Methods; 51 patients received bilateral (UG) block: (ESP) group (n = 25), and (TAP) group (n = 26), Using the same dosage and volume of local anesthetics. Pain intensity using VAS score at first 30 minutes, and then at 2, 4, 6, 8, 12, 16, 20, and 24 hours postoperatively, IV boluses of pethidine (rescue analgesic) were administrated if VAS ≥ 4, The primary outcome considered as the duration of effective analgesia for each block. While the Secondary outcome measured the over-all pethidine usage, patient satisfaction and the incidence of harmful effects from the techniques. RESULTS; VAS score in the ESP group was significantly lower at 8hrs and 12hrs. There was significant prolonged time to the first analgesic dose in ESP group (9.16±1.07 hours) than TAP group (7.65±0.75 hours). Also, a significant reduction in to the overall pethidine intake in 24 hours (110.40±12.74mg.) in the ESP group. Conclusion: Especially in comparison to the TAP block, the ESP block enables more reliable alleviation of pain, longer analgesic period, extends the time to initial analgesic requirements, reduces pethidine usage.

We obtain two matrix inequalities on the off-diagonal block of 2×2 block positive semidefinite matrices and the geometric mean of its diagonal blocks. They improve some results of Lee [Linear Algeb...

The sum of squares of the degrees of the vertices of a graph G is called the first Zagreb index of a graph G and the first Zagreb coindex of a graph G is defined as . In this paper, expressions for...

A basic problem of the constant dimension subspace coding is to determine the maximal possible size Aq(n,d,k) of a set of k-dimensional subspaces in $\mathbf {F}_{q}^{n}$ such that the subspace distance satisfies $\text {dis}(U,V) =2k-2 \dim (U \cap V) \geq d$ for any two different subspaces U and V in this set. We propose two constructions of constant dimension subspace codes that can insert flexibly into the generalized parallel linkage construction. In our constructions matrix blocks from small constant dimension codes and rank metric codes play important roles. Through a well-arranged combination for the matrix blocks, more than 120 new constant dimension subspace codes of distance 4, 6, 8 better than previously best known codes are constructed.

A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneer results obtained by Verghese, Van Dooren and Kailath in 1979-80, which were the first proving results of this type under different nonequivalent conditions. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices.

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

Introduction The erector spinae plane (ESP) block and thoracolumbar interfascial plane (TLIP) block were two novel plane blocks. The purpose of this study was to investigate TLIP block and ESP block on the effect of analgesic and opioid consumption in lumbar spine fusion surgery in the perioperative period. Methods Three hundred and four patients who suffered lumbar spine fusion were included and randomly divided into three groups: a control group (n = 102), an ESP block group (n = 100), and a TLIP block group (n = 102). We recorded the numerical rating scale (NRS) pain at movement and static during the postoperative 48 h, opioid consumption, additional analgesic requirement, frequency of patient-controlled analgesia (PCA) compressions, Bruggemann Comfort Scale (BCS) score, side effects, duration of hospital stay, and the life quality score (LQS) after operation at 6 months. Results The patients in the ESP block group have better analgesia during 12–48 h postoperative time at static state, a lower frequency of PCA compressions at 24–48 h after surgery, and the opioid consumption in the PCA (sufentanil) were less than those in the TLIP block group (P

Background Surgical repair of congenital inguinal hernia results in a significant postoperative discomfort and pain. The aim of the current study was to evaluate the pre-emptive analgesic efficacy of transversalis fascia plane (TFP) block after pediatric inguinal herniorrhaphy. Methods In this randomized, double-blinded, controlled, superiority study, 44 patients aged 12 to 60 months, undergoing unilateral inguinal herniorrhaphy were enrolled. Four patients were excluded and the remaining were allocated into the control group and TFP block group. In TFP block group 0.4 mL/kg bupivacaine 0.25% was instilled in the plane between transversus abdominis and transversalis fascia, while in control group 0.9 saline was used instead of bupivacaine. The primary outcome measure was the total dose of paracetamol consumed during the first postoperative 12 hours (h). The secondary outcome measures were the postoperative Face, Leg, Activity, Cry, Consolability (FLACC) pain score, time to first rescue analgesia, number of children required additional postoperative analgesics and parents' satisfaction Likert scale. Results During the first postoperative 12 h, the median paracetamol consumption was significantly lower in TFP than the control group and FLACC pain scores were significantly lower all over the study time in TFP block group with higher parents' satisfaction score than the control group. The time to first rescue analgesia was significantly longer and the number of patients who required additional analgesics were significantly lower in TFP block group than the control group. Conclusions The use of TFP block decreases postoperative analgesic consumption and postoperative pain intensity after pediatric inguinal herniorrhaphy.

Let G be a finite group. We prove that if the set of degrees of characters in the principal p-block of G has size at most 2 then G is p-solvable, and G / O p ′ ( G ) has a metabelian normal Sylow p-subgroup. The general question of proving that if an arbitrary p-block has two degrees then their defect groups are metabelian remains open.

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

Subset selection for matrices is the task of extracting a column sub-matrix from a given matrix $B\in\mathbb{R}^{n\times m}$ with $m>n$ such that the pseudoinverse of the sampled matrix has as small Frobenius or spectral norm as possible. In this paper, we consider a more general problem of subset selection for matrices that allows a block to be fixed at the beginning. Under this setting, we provide a deterministic method for selecting a column sub-matrix from $B$. We also present a bound for both the Frobenius and spectral norms of the pseudoinverse of the sampled matrix, showing that the bound is asymptotically optimal. The main technology for proving this result is the interlacing families of polynomials developed by Marcus, Spielman, and Srivastava. This idea also results in a deterministic greedy selection algorithm that produces the sub-matrix promised by our result.

Low-dose spinal saddle anaesthesia is a safe and reliable anaesthesia technique in outpatient perianal surgery, which provides a reliable but restricted block with good surgical conditions. Chloroprocaine was re-introduced without antioxidant and preservatives, which can be used safely in spinal anesthesia. Fentanyl is an opiod agonist, acts on mu receptor and of addition of fentanyl to chlorprocaine appears to lengthen the regression to L1 dermatome while minimally lengthening duration of block. This study was done to compare the effects of two different doses of fentanyl as an adjuvant to 1% Chloroprocaine under saddle block for perianal surgeries, and duration of stay in hospital.Materials and Methods: The study was a randomized controlled double blinded trial conducted on 45 patients aged between 18 – 65 years (ASA 1 and 2) posted for perianal surgeries under spinal anaesthesia in private hospitals in and around Dakshina Kannada from January 2018 to June 2021. The groups are assigned as follows: Group A (n = 15) received1% Chloroprocaine 20 mg plus 0.5 ml of normal saline intrathecally. Group B (n = 15) received1% Chloroprocaine 20mg plus fentanyl 15 µg (0.3ml) + 0.2ml of normal saline intrathecally.Group C (n = 15) received1% Chloroprocaine 20 mg plus fentanyl 25 µg (0.5ml) intrathecally. The sensory block, modified bromage scale, post operatively analgesia duration and the rescue analgesia patient’s and surgeon’s satisfaction score, side effects were noted.Observations and Results: There was no significant difference in the maximum Bromage score for the motor block it was 11.57 seconds in Group and Group B and 11. 11.6 in group C . NRS score that denotes the quality of analgesia was higher in group A indicating that the satisfaction was lesser .In the group A the time was the longest followed by group B and group C had the least two segment sensory regression time and was statistically significant with a p value< 0.05. In our study in the group A the time for motor regression to baseline was the highest followed by group C and group B had the least time for the motor regression to baseline and was statistically significant with a p value < 0.05.group A required the highest number of analgesia doses followed by group B and group C required equal number of analgesia and was statistically significant with a p value < 0.05. Group A had the least patient and surgeon satisfaction scores followed by group B and group C had the maximum time and was statistically significant with a p value < 0.05.Conclusion: Group B had better hemodynamic parameters than group A and group C . Group C was associated with a lot of adverse event intra op like hypotension and group A had a higher need of analgesic and faster sensory loss Hence, considering the nature and duration of the procedure, low dose saddle block anaesthesia with 2-Chloroprocaine would be a suitable choice and we evaluated the drug in varying doses to study the clinical response of varying doses, for procedures lasting for small duration 20 mg is sufficient, the addition of fentanyl may be adjusted based on the expected time of surgery.

Summary The pericapsular nerve group (PENG) block is a novel regional anaesthesia technique that aims to provide hip analgesia with preservation of motor function, although evidence is currently lacking. In this single‐centre, observer‐masked, randomised controlled trial, patients undergoing total hip arthroplasty received pericapsular nerve group block or no block (control group). Primary outcome measure was maximum pain scores (0–10 numeric rating scale) measured in the first 48 h after surgery. Secondary outcomes included postoperative opioid consumption; patient mobilisation assessments; and length of hospital stay. Sixty patients were randomly allocated equally between groups. The maximum pain score of patients receiving the pericapsular nerve group block was significantly lower than in the control group at all time‐points, with a median (IQR [range]) of 2.5 (2.0–3.7 [0–7]) vs. 5.5 (5.0–7.0 [2–8]) at 12 h; 3 (2.0–4.0 [0–7]) vs. 6 (5.0–6.0 [2–8]) at 24 h; and 2.0 (2.0–4.0 [0–5]) vs. 3.0 (2.0–4.7 [0–6]) at 48 h; all p

In this paper, we prove that there is an equivariant basic Morita equivalence between an inertial block and its Brauer correspondent in the normalizer of a defect group of the block.

Objective: To find the effectiveness of ultrasound guided modified pectoral nerve block (PECS II) versus conventional analgesics for post-operative pain relief in women undergoing modified radical mastectomy. Study Design: Quasi experimental study. Place and Duration of Study: Department of Anaesthesia, Pak Emirates Military Hospital Rawalpindi Pakistan, from May 2018 to Oct 2019. Methodology: A total of 74 adult female patients scheduled for elective unilateral modified radical mastectomy under general anaesthesia were randomized into two groups, pectoral nerve block II (P) group (n=37) and control (C) group (n=37). An ultrasound-guided pectoral nerve block II block was performed using 30 ml of 0.25% Bupivacaine in pectoral nerve block II group after induction of general anaesthesia. In control group (C), patients received only general anaesthesia. Primary outcome measure was opioid consumption in first 24 hours, and the secondary outcome was pain at the breast and axillary region measured using the numerical rating scale (NRS) in first 24 hours at fixed intervals after surgery. Patient satisfaction was evaluated upon discharge using a 4-point scale. Results: Nalbuphine consumption was significantly reduced in pectoral nerve block group as compared to control group. Patients in pectoral nerve block II group experienced much less pain because their numerical rating scale was lower than the control group in postoperative period. Patient satisfaction was found to be high in pectoral nerve block II group. Conclusion: The pectoral nerve block II block is a simple block which provides excellent analgesia for modified radical mastectomy. It can be used for balanced anaesthesia.......

We consider 1d random Hermitian $$N\times N$$ block band matrices consisting of $$W\times W$$ random Gaussian blocks (parametrized by $$j,k \in \Lambda =[1,n]\cap \mathbb {Z}$$ , $$N=nW$$ ) with a fixed entry’s variance $$J_{jk}=W^{-1}(\delta _{j,k}+\beta \Delta _{j,k})$$ in each block. Considering the limit $$W, n\rightarrow \infty $$ , we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as $$W\gg \sqrt{N}$$ , is determined by the Wigner–Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in Shcherbina and Shcherbina (J Stat Phys 172:627–664, 2018).